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Q-analysis
Q-analysis is a mathematical framework to describe and analyze set systems, or equivalently simplicial complexes. This idea was first introduced by Ronald Atkin in the early 1970s. Atkin was a British mathematician teaching at the University of Essex. Crediting the inspiration of his idea to Clifford Dowker’s paper (Homology Groups of Relations, Annals of Mathematics, 1952), he became interested in the algebra of relations in social structures. He tried to explain his idea in both mathematical and also accessible forms to both technical and general audience. His main ideas are reflected in The Mathematical Structure of Human Affairs (1974). That book covers the key ideas in q-analysis and its application to a wide range of examples, like analyzing game of chess, urban structures, politics at university, people and complexes, works of abstract art, and to physics. He contended that q-analysis can be considered as a powerful generalized method wherever we are dealing with relationships among sets.[1]
Description
A simplex of n vertices can be represented as a polyhedron in n − 1 dimensions, so that for example a triangle of three vertices can be drawn on a plane of two dimensions and is accordingly called a 2-simplex. When simplices share vertices, the intersections of their vertex sets are themselves simplices of equal or lower dimension. For example, two triangles with two vertices in common share not only the two 0-simplex vertices but the 1-simplex line between them. The triangles are said to be both 1- and 0-connected because they share 1- and 0-dimensional faces.
Q-analysis of a simplicial complex consists in stepping through all q up to the dimension of the largest simplex and constructing for each q a graph of the simplices that are q-connected at each level, and in particular, determining how many connected components are present for each q.[2]
Q-analysis can thus provide a rich summary of (literally) multi-faceted relationships between entities.
Applications
• Analysis of large-scale systems structure
• Analysis of social network
• Decision making
See also
• Systems theory
• Living systems theory
Notes
1. Jacky Legrand. How far can Q-analysis go into social systems understanding?. Fifth European Systems Science Congress, 2002.
• Atkin, R. (1974). Mathematical Structure in Human Affairs. London, Heinemann.
References
• Atkin, R. (1972). From cohomology in physics to q-connectivity in social science. International Journal of Man-Machines Studies vol. 4, 139–167.
• Atkin, R. (1974). Mathematical Structure in Human Affairs. London, Heinemann.
• Atkin, R. (1976). An algebra for patterns on a complex II. International Journal of Man-Machines Studies vol. 8, 483–498.
• Atkin, R. (1977). Combinatorial Connectivities in Social Systems. Basel, Birkhäuser Verlag.
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Q-category
In mathematics, a Q-category or almost quotient category[1] is a category that is a "milder version of a Grothendieck site."[2] A Q-category is a coreflective subcategory.[1] The Q stands for a quotient.
The concept of Q-categories was introduced by Alexander Rosenberg in 1988.[2] The motivation for the notion was its use in noncommutative algebraic geometry; in this formalism, noncommutative spaces are defined as sheaves on Q-categories.
Definition
A Q-category is defined by the formula[1]
$\mathbb {A} :(u^{*}\dashv u_{*}):{\bar {A}}{\stackrel {\overset {u^{*}}{\leftarrow }}{\underset {u_{*}}{\to }}}A$ :(u^{*}\dashv u_{*}):{\bar {A}}{\stackrel {\overset {u^{*}}{\leftarrow }}{\underset {u_{*}}{\to }}}A}
where $u^{*}$ is the left adjoint in a pair of adjoint functors and is a full and faithful functor.
Examples
• The category of presheaves over any Q-category is itself a Q-category.[1]
• For any category, one can define the Q-category of cones.[1]
• There is a Q-category of sieves.[1]
References
1. Škoda, Zoran; Schreiber, Urs; Mrđen, Rafael; Fritz, Tobias (14 September 2017). "Q-category". nLab. Retrieved 25 March 2023.
2. Kontsevich & Rosenberg 2004a, § 1.
• Kontsevich, Maxim; Rosenberg, Alexander (2004a). "Noncommutative spaces" (PDF). ncatlab.org. Retrieved 25 March 2023.{{cite web}}: CS1 maint: url-status (link)
• Alexander Rosenberg, Q-categories, sheaves and localization, (in Russian) Seminar on supermanifolds 25, Leites ed. Stockholms Universitet 1988.
Further reading
• Kontsevich, Maxim; Rosenberg, Alexander (2004b). "Noncommutative stacks". ncatlab.org. Retrieved 25 March 2023.{{cite web}}: CS1 maint: url-status (link)
• Brzezinski, Tomasz (29 October 2007). Brzeziński, Tomasz; Pardo, José Luis Gómez; Shestakov, Ivan; Smith, Patrick F. (eds.). Notes on formal smoothness. Modules and Comodules. arXiv:0710.5527.
• Lawvere, F. William (2007). "Axiomatic Cohesion" (PDF). Theory and Applications of Categories. 19 (3): 41–49.
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q-Gaussian process
q-Gaussian processes are deformations of the usual Gaussian distribution. There are several different versions of this; here we treat a multivariate deformation, also addressed as q-Gaussian process, arising from free probability theory and corresponding to deformations of the canonical commutation relations. For other deformations of Gaussian distributions, see q-Gaussian distribution and Gaussian q-distribution.
History
The q-Gaussian process was formally introduced in a paper by Frisch and Bourret[1] under the name of parastochastics, and also later by Greenberg[2] as an example of infinite statistics. It was mathematically established and investigated in papers by Bozejko and Speicher[3] and by Bozejko, Kümmerer, and Speicher[4] in the context of non-commutative probability.
It is given as the distribution of sums of creation and annihilation operators in a q-deformed Fock space. The calculation of moments of those operators is given by a q-deformed version of a Wick formula or Isserlis formula. The specification of a special covariance in the underlying Hilbert space leads to the q-Brownian motion,[4] a special non-commutative version of classical Brownian motion.
q-Fock space
In the following $q\in [-1,1]$ is fixed. Consider a Hilbert space ${\mathcal {H}}$. On the algebraic full Fock space
${\mathcal {F}}_{\text{alg}}({\mathcal {H}})=\bigoplus _{n\geq 0}{\mathcal {H}}^{\otimes n},$
where ${\mathcal {H}}^{0}=\mathbb {C} \Omega $ with a norm one vector $\Omega $, called vacuum, we define a q-deformed inner product as follows:
$\langle h_{1}\otimes \cdots \otimes h_{n},g_{1}\otimes \cdots \otimes g_{m}\rangle _{q}=\delta _{nm}\sum _{\sigma \in S_{n}}\prod _{r=1}^{n}\langle h_{r},g_{\sigma (r)}\rangle q^{i(\sigma )},$
where $i(\sigma )=\#\{(k,\ell )\mid 1\leq k<\ell \leq n;\sigma (k)>\sigma (\ell )\}$ is the number of inversions of $\sigma \in S_{n}$.
The q-Fock space[5] is then defined as the completion of the algebraic full Fock space with respect to this inner product
${\mathcal {F}}_{q}({\mathcal {H}})={\overline {\bigoplus _{n\geq 0}{\mathcal {H}}^{\otimes n}}}^{\langle \cdot ,\cdot \rangle _{q}}.$
For $-1<q<1$ the q-inner product is strictly positive.[3][6] For $q=1$ and $q=-1$ it is positive, but has a kernel, which leads in these cases to the symmetric and anti-symmetric Fock spaces, respectively.
For $h\in {\mathcal {H}}$ we define the q-creation operator $a^{*}(h)$, given by
$a^{*}(h)\Omega =h,\qquad a^{*}(h)h_{1}\otimes \cdots \otimes h_{n}=h\otimes h_{1}\otimes \cdots \otimes h_{n}.$
Its adjoint (with respect to the q-inner product), the q-annihilation operator $a(h)$, is given by
$a(h)\Omega =0,\qquad a(h)h_{1}\otimes \cdots \otimes h_{n}=\sum _{r=1}^{n}q^{r-1}\langle h,h_{r}\rangle h_{1}\otimes \cdots \otimes h_{r-1}\otimes h_{r+1}\otimes \cdots \otimes h_{n}.$
q-commutation relations
Those operators satisfy the q-commutation relations[7]
$a(f)a^{*}(g)-qa^{*}(g)a(f)=\langle f,g\rangle \cdot 1\qquad (f,g\in {\mathcal {H}}).$
For $q=1$, $q=0$, and $q=-1$ this reduces to the CCR-relations, the Cuntz relations, and the CAR-relations, respectively. With the exception of the case $q=1,$ the operators $a^{*}(f)$ are bounded.
q-Gaussian elements and definition of multivariate q-Gaussian distribution (q-Gaussian process)
Operators of the form $s_{q}(h)={a(h)+a^{*}(h)}$ for $h\in {\mathcal {H}}$ are called q-Gaussian[5] (or q-semicircular[8]) elements.
On ${\mathcal {F}}_{q}({\mathcal {H}})$ we consider the vacuum expectation state $\tau (T)=\langle \Omega ,T\Omega \rangle $, for $T\in {\mathcal {B}}({\mathcal {F}}({\mathcal {H}}))$.
The (multivariate) q-Gaussian distribution or q-Gaussian process[4][9] is defined as the non commutative distribution of a collection of q-Gaussians with respect to the vacuum expectation state. For $h_{1},\dots ,h_{p}\in {\mathcal {H}}$ the joint distribution of $s_{q}(h_{1}),\dots ,s_{q}(h_{p})$ with respect to $\tau $ can be described in the following way,:[1][3] for any $i\{1,\dots ,k\}\rightarrow \{1,\dots ,p\}$ we have
$\tau \left(s_{q}(h_{i(1)})\cdots s_{q}(h_{i(k)})\right)=\sum _{\pi \in {\mathcal {P}}_{2}(k)}q^{cr(\pi )}\prod _{(r,s)\in \pi }\langle h_{i(r)},h_{i(s)}\rangle ,$
where $cr(\pi )$ denotes the number of crossings of the pair-partition $\pi $. This is a q-deformed version of the Wick/Isserlis formula.
q-Gaussian distribution in the one-dimensional case
For p = 1, the q-Gaussian distribution is a probability measure on the interval $[-2/{\sqrt {1-q}},2/{\sqrt {1-q}}]$, with analytic formulas for its density.[10] For the special cases $q=1$, $q=0$, and $q=-1$, this reduces to the classical Gaussian distribution, the Wigner semicircle distribution, and the symmetric Bernoulli distribution on $\pm 1$. The determination of the density follows from old results[11] on corresponding orthogonal polynomials.
Operator algebraic questions
The von Neumann algebra generated by $s_{q}(h_{i})$, for $h_{i}$ running through an orthonormal system $(h_{i})_{i\in I}$ of vectors in ${\mathcal {H}}$, reduces for $q=0$ to the famous free group factors $L(F_{\vert I\vert })$. Understanding the structure of those von Neumann algebras for general q has been a source of many investigations.[12] It is now known, by work of Guionnet and Shlyakhtenko,[13] that at least for finite I and for small values of q, the von Neumann algebra is isomorphic to the corresponding free group factor.
References
1. Frisch, U.; Bourret, R. (February 1970). "Parastochastics". Journal of Mathematical Physics. 11 (2): 364–390. Bibcode:1970JMP....11..364F. doi:10.1063/1.1665149.
2. Greenberg, O. W. (12 February 1990). "Example of infinite statistics". Physical Review Letters. 64 (7): 705–708. Bibcode:1990PhRvL..64..705G. doi:10.1103/PhysRevLett.64.705. PMID 10042057.
3. Bożejko, Marek; Speicher, Roland (April 1991). "An example of a generalized Brownian motion". Communications in Mathematical Physics. 137 (3): 519–531. Bibcode:1991CMaPh.137..519B. doi:10.1007/BF02100275. S2CID 123190397.
4. Bożejko, M.; Kümmerer, B.; Speicher, R. (1 April 1997). "q-Gaussian Processes: Non-commutative and Classical Aspects". Communications in Mathematical Physics. 185 (1): 129–154. arXiv:funct-an/9604010. Bibcode:1997CMaPh.185..129B. doi:10.1007/s002200050084. S2CID 2993071.
5. Effros, Edward G.; Popa, Mihai (22 July 2003). "Feynman diagrams and Wick products associated with q-Fock space". Proceedings of the National Academy of Sciences. 100 (15): 8629–8633. arXiv:math/0303045. Bibcode:2003PNAS..100.8629E. doi:10.1073/pnas.1531460100. PMC 166362. PMID 12857947.
6. Zagier, Don (June 1992). "Realizability of a model in infinite statistics". Communications in Mathematical Physics. 147 (1): 199–210. Bibcode:1992CMaPh.147..199Z. CiteSeerX 10.1.1.468.966. doi:10.1007/BF02099535. S2CID 53385666.
7. Kennedy, Matthew; Nica, Alexandru (9 September 2011). "Exactness of the Fock Space Representation of the q-Commutation Relations". Communications in Mathematical Physics. 308 (1): 115–132. arXiv:1009.0508. Bibcode:2011CMaPh.308..115K. doi:10.1007/s00220-011-1323-9. S2CID 119124507.
8. Vergès, Matthieu Josuat (20 November 2018). "Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps". Canadian Journal of Mathematics. 65 (4): 863–878. arXiv:1203.3157. doi:10.4153/CJM-2012-042-9. S2CID 2215028.
9. Bryc, Włodzimierz; Wang, Yizao (24 February 2016). "The local structure of q-Gaussian processes". arXiv:1511.06667. {{cite journal}}: Cite journal requires |journal= (help)
10. Leeuwen, Hans van; Maassen, Hans (September 1995). "A q deformation of the Gauss distribution". Journal of Mathematical Physics. 36 (9): 4743–4756. Bibcode:1995JMP....36.4743V. doi:10.1063/1.530917. hdl:2066/141604.
11. Szegö, G (1926). "Ein Beitrag zur Theorie der Thetafunktionen" [A contribution to the theory of theta functions]. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-Math. Klasse (in German): 242–252.
12. Wasilewski, Mateusz (24 February 2020). "A simple proof of the complete metric approximation property for q-Gaussian algebras". arXiv:1907.00730. {{cite journal}}: Cite journal requires |journal= (help)
13. Guionnet, A.; Shlyakhtenko, D. (13 November 2013). "Free monotone transport". Inventiones Mathematicae. 197 (3): 613–661. arXiv:1204.2182. doi:10.1007/s00222-013-0493-9. S2CID 16882208.
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Quantum calculus
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula
$q=e^{ih}=e^{2\pi i\hbar }$
where $ \hbar ={\frac {h}{2\pi }}$ is the reduced Planck constant.
Differentiation
In the q-calculus and h-calculus, differentials of functions are defined as
$d_{q}(f(x))=f(qx)-f(x)$
and
$d_{h}(f(x))=f(x+h)-f(x)$
respectively. Derivatives of functions are then defined as fractions by the q-derivative
$D_{q}(f(x))={\frac {d_{q}(f(x))}{d_{q}(x)}}={\frac {f(qx)-f(x)}{(q-1)x}}$
and by
$D_{h}(f(x))={\frac {d_{h}(f(x))}{d_{h}(x)}}={\frac {f(x+h)-f(x)}{h}}$
In the limit, as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.
Integration
q-integral
A function F(x) is a q-antiderivative of f(x) if DqF(x) = f(x). The q-antiderivative (or q-integral) is denoted by $ \int f(x)\,d_{q}x$ and an expression for F(x) can be found from the formula $ \int f(x)\,d_{q}x=(1-q)\sum _{j=0}^{\infty }xq^{j}f(xq^{j})$ which is called the Jackson integral of f(x). For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if |f(x)xα| is bounded on the interval (0, A] for some 0 ≤ α < 1.
The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points qj, with the jump at the point qj being qj. If we call this step function gq(t) then dgq(t) = dqt.[1]
h-integral
A function F(x) is an h-antiderivative of f(x) if DhF(x) = f(x). The h-antiderivative (or h-integral) is denoted by $ \int f(x)\,d_{h}x$. If a and b differ by an integer multiple of h then the definite integral $ \int _{a}^{b}f(x)\,d_{h}x$ is given by a Riemann sum of f(x) on the interval [a, b] partitioned into subintervals of width h.
Example
The derivative of the function $x^{n}$ (for some positive integer $n$) in the classical calculus is $nx^{n-1}$. The corresponding expressions in q-calculus and h-calculus are
$D_{q}(x^{n})={\frac {q^{n}-1}{q-1}}x^{n-1}=[n]_{q}\ x^{n-1}$
with the q-bracket
$[n]_{q}={\frac {q^{n}-1}{q-1}}$
and
$D_{h}(x^{n})=nx^{n-1}+{\frac {n(n-1)}{2}}hx^{n-2}+\cdots +h^{n-1}$
respectively. The expression $[n]_{q}x^{n-1}$ is then the q-calculus analogue of the simple power rule for positive integral powers. In this sense, the function $x^{n}$ is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of $x^{n}$ is instead the falling factorial, $(x)_{n}:=x(x-1)\cdots (x-n+1).$ One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine.
History
The h-calculus is just the calculus of finite differences, which had been studied by George Boole and others, and has proven useful in a number of fields, among them combinatorics and fluid mechanics. The q-calculus, while dating in a sense back to Leonhard Euler and Carl Gustav Jacobi, is only recently beginning to see more usefulness in quantum mechanics, having an intimate connection with commutativity relations and Lie algebra.
See also
• Noncommutative geometry
• Quantum differential calculus
• Time scale calculus
• q-analog
• Basic hypergeometric series
• Quantum dilogarithm
Further reading
• George Gasper, Mizan Rahman, Basic Hypergeometric Series, 2nd ed, Cambridge University Press (2004), ISBN 978-0-511-52625-1, doi:10.1017/CBO9780511526251
References
1. Abreu, Luis Daniel (2006). "Functions q-Orthogonal with Respect to Their Own Zeros" (PDF). Proceedings of the American Mathematical Society. 134 (9): 2695–2702. doi:10.1090/S0002-9939-06-08285-2. JSTOR 4098119.
• Jackson, F. H. (1908). "On q-functions and a certain difference operator". Transactions of the Royal Society of Edinburgh. 46 (2): 253–281. doi:10.1017/S0080456800002751. S2CID 123927312.
• Exton, H. (1983). q-Hypergeometric Functions and Applications. New York: Halstead Press. ISBN 0-85312-491-4.
• Kac, Victor; Cheung, Pokman (2002). Quantum calculus. Universitext. Springer-Verlag. ISBN 0-387-95341-8.
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
• Quantum biology
• Quantum chemistry
• Quantum chaos
• Quantum cosmology
• Quantum differential calculus
• Quantum dynamics
• Quantum geometry
• Quantum measurement problem
• Quantum mind
• Quantum stochastic calculus
• Quantum spacetime
Technology
• Quantum algorithms
• Quantum amplifier
• Quantum bus
• Quantum cellular automata
• Quantum finite automata
• Quantum channel
• Quantum circuit
• Quantum complexity theory
• Quantum computing
• Timeline
• Quantum cryptography
• Quantum electronics
• Quantum error correction
• Quantum imaging
• Quantum image processing
• Quantum information
• Quantum key distribution
• Quantum logic
• Quantum logic gates
• Quantum machine
• Quantum machine learning
• Quantum metamaterial
• Quantum metrology
• Quantum network
• Quantum neural network
• Quantum optics
• Quantum programming
• Quantum sensing
• Quantum simulator
• Quantum teleportation
Extensions
• Casimir effect
• Quantum statistical mechanics
• Quantum field theory
• History
• Quantum gravity
• Relativistic quantum mechanics
Related
• Schrödinger's cat
• in popular culture
• EPR paradox
• Quantum mysticism
• Category
• Physics portal
• Commons
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q-difference polynomial
In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.
Definition
The q-difference polynomials satisfy the relation
$\left({\frac {d}{dz}}\right)_{q}p_{n}(z)={\frac {p_{n}(qz)-p_{n}(z)}{qz-z}}={\frac {q^{n}-1}{q-1}}p_{n-1}(z)=[n]_{q}p_{n-1}(z)$
where the derivative symbol on the left is the q-derivative. In the limit of $q\to 1$, this becomes the definition of the Appell polynomials:
${\frac {d}{dz}}p_{n}(z)=np_{n-1}(z).$
Generating function
The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
$A(w)e_{q}(zw)=\sum _{n=0}^{\infty }{\frac {p_{n}(z)}{[n]_{q}!}}w^{n}$
where $e_{q}(t)$ is the q-exponential:
$e_{q}(t)=\sum _{n=0}^{\infty }{\frac {t^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {t^{n}(1-q)^{n}}{(q;q)_{n}}}.$
Here, $[n]_{q}!$ is the q-factorial and
$(q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)$
is the q-Pochhammer symbol. The function $A(w)$ is arbitrary but assumed to have an expansion
$A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}{\mbox{ with }}a_{0}\neq 0.$
Any such $A(w)$ gives a sequence of q-difference polynomials.
References
• A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", Riv. Mat. Univ. Parma, 5 (1954) 325–337.
• Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. (Provides a very brief discussion of convergence.)
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q-expansion principle
In mathematics, the q-expansion principle states that a modular form f has coefficients in a module M if its q-expansion at enough cusps resembles the q-expansion of a modular form g with coefficients in M. It was introduced by Katz (1973, corollaries 1.6.2, 1.12.2).
References
• Katz, Nicholas M. (1973), "p-adic properties of modular schemes and modular forms", Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics, vol. 350, Berlin, New York: Springer-Verlag, pp. 69–190, doi:10.1007/978-3-540-37802-0_3, ISBN 978-3-540-06483-1, MR 0447119
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q-exponential
In combinatorial mathematics, a q-exponential is a q-analog of the exponential function, namely the eigenfunction of a q-derivative. There are many q-derivatives, for example, the classical q-derivative, the Askey-Wilson operator, etc. Therefore, unlike the classical exponentials, q-exponentials are not unique. For example, $e_{q}(z)$ is the q-exponential corresponding to the classical q-derivative while ${\mathcal {E}}_{q}(z)$ are eigenfunctions of the Askey-Wilson operators.
Not to be confused with the Tsallis q-exponential.
Definition
The q-exponential $e_{q}(z)$ is defined as
$e_{q}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {z^{n}(1-q)^{n}}{(q;q)_{n}}}=\sum _{n=0}^{\infty }z^{n}{\frac {(1-q)^{n}}{(1-q^{n})(1-q^{n-1})\cdots (1-q)}}$
where $[n]!_{q}$ is the q-factorial and
$(q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)$
is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property
$\left({\frac {d}{dz}}\right)_{q}e_{q}(z)=e_{q}(z)$
where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial
$\left({\frac {d}{dz}}\right)_{q}z^{n}=z^{n-1}{\frac {1-q^{n}}{1-q}}=[n]_{q}z^{n-1}.$
Here, $[n]_{q}$ is the q-bracket. For other definitions of the q-exponential function, see Exton (1983), Ismail & Zhang (1994), Suslov (2003) harvtxt error: no target: CITEREFSuslov2003 (help) and Cieśliński (2011).
Properties
For real $q>1$, the function $e_{q}(z)$ is an entire function of $z$. For $q<1$, $e_{q}(z)$ is regular in the disk $|z|<1/(1-q)$.
Note the inverse, $~e_{q}(z)~e_{1/q}(-z)=1$.
Addition Formula
The analogue of $\exp(x)\exp(y)=\exp(x+y)$ does not hold for real numbers $x$ and $y$. However, if these are operators satisfying the commutation relation $xy=qyx$, then $e_{q}(x)e_{q}(y)=e_{q}(x+y)$ holds true.[1]
Relations
For $-1<q<1$, a function that is closely related is $E_{q}(z).$ It is a special case of the basic hypergeometric series,
$E_{q}(z)=\;_{1}\phi _{1}\left({\scriptstyle {0 \atop 0}}\,;\,z\right)=\sum _{n=0}^{\infty }{\frac {q^{\binom {n}{2}}(-z)^{n}}{(q;q)_{n}}}=\prod _{n=0}^{\infty }(1-q^{n}z)=(z;q)_{\infty }.$
Clearly,
$\lim _{q\to 1}E_{q}\left(z(1-q)\right)=\lim _{q\to 1}\sum _{n=0}^{\infty }{\frac {q^{\binom {n}{2}}(1-q)^{n}}{(q;q)_{n}}}(-z)^{n}=e^{-z}.~$
Relation with Dilogarithm
$e_{q}(x)$ has the following infinite product representation:
$e_{q}(x)=\left(\prod _{k=0}^{\infty }(1-q^{k}(1-q)x)\right)^{-1}.$
On the other hand, $\log(1-x)=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}$ holds. When $|q|<1$,
$\log e_{q}(x)=-\sum _{k=0}^{\infty }\log(1-q^{k}(1-q)x)=\sum _{k=0}^{\infty }\sum _{n=1}^{\infty }{\frac {(q^{k}(1-q)x)^{n}}{n}}=\sum _{n=1}^{\infty }{\frac {((1-q)x)^{n}}{(1-q^{n})n}}={\frac {1}{1-q}}\sum _{n=1}^{\infty }{\frac {((1-q)x)^{n}}{[n]_{q}n}}.$
By taking the limit $q\to 1$,
$\lim _{q\to 1}(1-q)\log e_{q}(x/(1-q))=\mathrm {Li} _{2}(x),$
where $\mathrm {Li} _{2}(x)$ is the dilogarithm.
In physics
Main article: Quantum dilogarithm
The Q-exponential function is also known as the quantum dilogarithm.[2][3]
References
1. Kac, V.; Cheung, P. (2011). Quantum Calculus. Springer. p. 31. ISBN 978-1461300724.
2. Zudilin, Wadim (14 March 2006). "Quantum dilogarithm" (PDF). wain.mi.ras.ru. Retrieved 16 July 2021.{{cite web}}: CS1 maint: url-status (link)
3. Faddeev, L.d.; Kashaev, R.m. (1994-02-20). "Quantum dilogarithm". Modern Physics Letters A. 09 (5): 427–434. arXiv:hep-th/9310070. Bibcode:1994MPLA....9..427F. doi:10.1142/S0217732394000447. ISSN 0217-7323. S2CID 119124642.
• Cieśliński, Jan L. (2011). "Improved q-exponential and q-trigonometric functions". Applied Mathematics Letters. 24 (12): 2110–2114. arXiv:1006.5652. doi:10.1016/j.aml.2011.06.009. S2CID 205496812.
• Exton, Harold (1983). q-Hypergeometric Functions and Applications. New York: Halstead Press, Chichester: Ellis Horwood. ISBN 0853124914.
• Gasper, George; Rahman, Mizan Rahman (2004). Basic Hypergeometric Series. Cambridge University Press. ISBN 0521833574.
• Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press. doi:10.1017/CBO9781107325982. ISBN 9780521782012.
• Ismail, Mourad E. H.; Zhang, Ruiming (1994). "Diagonalization of certain integral operators". Advances in Mathematics. 108 (1): 1–33. doi:10.1006/aima.1994.1077.
• Ismail, Mourad E. H.; Rahman, Mizan; Zhang, Ruiming (1996). "Diagonalization of certain integral operators II". Journal of Computational and Applied Mathematics. 68 (1–2): 163–196. doi:10.1016/0377-0427(95)00263-4.
• Jackson, F. H. (1909). "On q-functions and a certain difference operator". Transactions of the Royal Society of Edinburgh. 46 (2): 253–281. doi:10.1017/S0080456800002751. S2CID 123927312.
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Basic hypergeometric series
In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.
The basic hypergeometric series ${}_{2}\phi _{1}(q^{\alpha },q^{\beta };q^{\gamma };q,x)$ was first considered by Eduard Heine (1846). It becomes the hypergeometric series $F(\alpha ,\beta ;\gamma ;x)$ ;\gamma ;x)} in the limit when base $q=1$.
Definition
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ. The unilateral basic hypergeometric series is defined as
$\;_{j}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{1+k-j}z^{n}$
where
$(a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}$
and
$(a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1})$
is the q-shifted factorial. The most important special case is when j = k + 1, when it becomes
$\;_{k+1}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k}&a_{k+1}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k+1};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}z^{n}.$
This series is called balanced if a1 ... ak + 1 = b1 ...bkq. This series is called well poised if a1q = a2b1 = ... = ak + 1bk, and very well poised if in addition a2 = −a3 = qa11/2. The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since
$\lim _{q\to 1}\;_{j}\phi _{k}\left[{\begin{matrix}q^{a_{1}}&q^{a_{2}}&\ldots &q^{a_{j}}\\q^{b_{1}}&q^{b_{2}}&\ldots &q^{b_{k}}\end{matrix}};q,(q-1)^{1+k-j}z\right]=\;_{j}F_{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};z\right]$
holds (Koekoek & Swarttouw (1996)).
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as
$\;_{j}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{k-j}z^{n}.$
The most important special case is when j = k, when it becomes
$\;_{k}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{k}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{k};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}z^{n}.$
The unilateral series can be obtained as a special case of the bilateral one by setting one of the b variables equal to q, at least when none of the a variables is a power of q, as all the terms with n < 0 then vanish.
Simple series
Some simple series expressions include
${\frac {z}{1-q}}\;_{2}\phi _{1}\left[{\begin{matrix}q\;q\\q^{2}\end{matrix}}\;;q,z\right]={\frac {z}{1-q}}+{\frac {z^{2}}{1-q^{2}}}+{\frac {z^{3}}{1-q^{3}}}+\ldots $
and
${\frac {z}{1-q^{1/2}}}\;_{2}\phi _{1}\left[{\begin{matrix}q\;q^{1/2}\\q^{3/2}\end{matrix}}\;;q,z\right]={\frac {z}{1-q^{1/2}}}+{\frac {z^{2}}{1-q^{3/2}}}+{\frac {z^{3}}{1-q^{5/2}}}+\ldots $
and
$\;_{2}\phi _{1}\left[{\begin{matrix}q\;-1\\-q\end{matrix}}\;;q,z\right]=1+{\frac {2z}{1+q}}+{\frac {2z^{2}}{1+q^{2}}}+{\frac {2z^{3}}{1+q^{3}}}+\ldots .$
The q-binomial theorem
The q-binomial theorem (first published in 1811 by Heinrich August Rothe)[1][2] states that
$\;_{1}\phi _{0}(a;q,z)={\frac {(az;q)_{\infty }}{(z;q)_{\infty }}}=\prod _{n=0}^{\infty }{\frac {1-aq^{n}z}{1-q^{n}z}}$
which follows by repeatedly applying the identity
$\;_{1}\phi _{0}(a;q,z)={\frac {1-az}{1-z}}\;_{1}\phi _{0}(a;q,qz).$
The special case of a = 0 is closely related to the q-exponential.
Cauchy binomial theorem
Cauchy binomial theorem is a special case of the q-binomial theorem.[3]
$\sum _{n=0}^{N}y^{n}q^{n(n+1)/2}{\begin{bmatrix}N\\n\end{bmatrix}}_{q}=\prod _{k=1}^{N}\left(1+yq^{k}\right)\qquad (|q|<1)$
Ramanujan's identity
Srinivasa Ramanujan gave the identity
$\;_{1}\psi _{1}\left[{\begin{matrix}a\\b\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a;q)_{n}}{(b;q)_{n}}}z^{n}={\frac {(b/a,q,q/az,az;q)_{\infty }}{(b,b/az,q/a,z;q)_{\infty }}}$
valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for $\;_{6}\psi _{6}$ have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as
$\sum _{n=-\infty }^{\infty }q^{n(n+1)/2}z^{n}=(q;q)_{\infty }\;(-1/z;q)_{\infty }\;(-zq;q)_{\infty }.$
Ken Ono gives a related formal power series[4]
$A(z;q){\stackrel {\rm {def}}{=}}{\frac {1}{1+z}}\sum _{n=0}^{\infty }{\frac {(z;q)_{n}}{(-zq;q)_{n}}}z^{n}=\sum _{n=0}^{\infty }(-1)^{n}z^{2n}q^{n^{2}}.$
Watson's contour integral
As an analogue of the Barnes integral for the hypergeometric series, Watson showed that
${}_{2}\phi _{1}(a,b;c;q,z)={\frac {-1}{2\pi i}}{\frac {(a,b;q)_{\infty }}{(q,c;q)_{\infty }}}\int _{-i\infty }^{i\infty }{\frac {(qq^{s},cq^{s};q)_{\infty }}{(aq^{s},bq^{s};q)_{\infty }}}{\frac {\pi (-z)^{s}}{\sin \pi s}}ds$
where the poles of $(aq^{s},bq^{s};q)_{\infty }$ lie to the left of the contour and the remaining poles lie to the right. There is a similar contour integral for r+1φr. This contour integral gives an analytic continuation of the basic hypergeometric function in z.
Matrix version
The basic hypergeometric matrix function can be defined as follows:
${}_{2}\phi _{1}(A,B;C;q,z):=\sum _{n=0}^{\infty }{\frac {(A;q)_{n}(B;q)_{n}}{(C;q)_{n}(q;q)_{n}}}z^{n},\quad (A;q)_{0}:=1,\quad (A;q)_{n}:=\prod _{k=0}^{n-1}(1-Aq^{k}).$
The ratio test shows that this matrix function is absolutely convergent.[5]
See also
• Dixon's identity
• Rogers–Ramanujan identities
Notes
1. Bressoud, D. M. (1981), "Some identities for terminating q-series", Mathematical Proceedings of the Cambridge Philosophical Society, 89 (2): 211–223, Bibcode:1981MPCPS..89..211B, doi:10.1017/S0305004100058114, MR 0600238.
2. Benaoum, H. B. (1998), "h-analogue of Newton's binomial formula", Journal of Physics A: Mathematical and General, 31 (46): L751–L754, arXiv:math-ph/9812011, Bibcode:1998JPhA...31L.751B, doi:10.1088/0305-4470/31/46/001, S2CID 119697596.
3. Wolfram Mathworld: Cauchy Binomial Theorem
4. Gwynneth H. Coogan and Ken Ono, A q-series identity and the Arithmetic of Hurwitz Zeta Functions, (2003) Proceedings of the American Mathematical Society 131, pp. 719–724
5. Ahmed Salem (2014) The basic Gauss hypergeometric matrix function and its matrix q-difference equation, Linear and Multilinear Algebra, 62:3, 347-361, DOI: 10.1080/03081087.2013.777437
References
• Andrews, G. E. (2010), "q-Hypergeometric and Related Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
• William Y. C. Chen and Amy Fu, Semi-Finite Forms of Bilateral Basic Hypergeometric Series (2004)
• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
• Sylvie Corteel and Jeremy Lovejoy, Frobenius Partitions and the Combinatorics of Ramanujan's $\,_{1}\psi _{1}$ Summation
• Fine, Nathan J. (1988), Basic hypergeometric series and applications, Mathematical Surveys and Monographs, vol. 27, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1524-3, MR 0956465
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
• Heine, Eduard (1846), "Über die Reihe $1+{\frac {(q^{\alpha }-1)(q^{\beta }-1)}{(q-1)(q^{\gamma }-1)}}x+{\frac {(q^{\alpha }-1)(q^{\alpha +1}-1)(q^{\beta }-1)(q^{\beta +1}-1)}{(q-1)(q^{2}-1)(q^{\gamma }-1)(q^{\gamma +1}-1)}}x^{2}+\cdots $", Journal für die reine und angewandte Mathematik, 32: 210–212
• Victor Kac, Pokman Cheung, Quantum calculus, Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8
• Koekoek, Roelof; Swarttouw, Rene F. (1996). The Askey scheme of orthogonal polynomials and its q-analogues (Report). Technical University Delft. no. 98-17.. Section 0.2
• Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions, Encyclopedia of Mathematics and its Applications, volume 71, Cambridge University Press.
• Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.
• Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin.
External links
• Weisstein, Eric W. "q-Hypergeometric Function". MathWorld.
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q-analog
In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.[1]
q-analogs are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit q → 1 is often formal, as q is often discrete-valued (for example, it may represent a prime power). q-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic dynamical systems. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket) and the modular group in particular. The connection passes through hyperbolic geometry and ergodic theory, where the elliptic integrals and modular forms play a prominent role; the q-series themselves are closely related to elliptic integrals.
q-analogs also appear in the study of quantum groups and in q-deformed superalgebras. The connection here is similar, in that much of string theory is set in the language of Riemann surfaces, resulting in connections to elliptic curves, which in turn relate to q-series.
"Classical" q-theory
Classical q-theory begins with the q-analogs of the nonnegative integers.[2] The equality
$\lim _{q\rightarrow 1}{\frac {1-q^{n}}{1-q}}=n$
suggests that we define the q-analog of n, also known as the q-bracket or q-number of n, to be
$[n]_{q}={\frac {1-q^{n}}{1-q}}=1+q+q^{2}+\ldots +q^{n-1}.$
By itself, the choice of this particular q-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use [n]q as the q-analog of n, one may define the q-analog of the factorial, known as the q-factorial, by
${\begin{aligned}{\big [}n]_{q}!&=[1]_{q}\cdot [2]_{q}\cdots [n-1]_{q}\cdot [n]_{q}\\[6pt]&={\frac {1-q}{1-q}}\cdot {\frac {1-q^{2}}{1-q}}\cdots {\frac {1-q^{n-1}}{1-q}}\cdot {\frac {1-q^{n}}{1-q}}\\[6pt]&=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1}).\end{aligned}}$
This q-analog appears naturally in several contexts. Notably, while n! counts the number of permutations of length n, [n]q! counts permutations while keeping track of the number of inversions. That is, if inv(w) denotes the number of inversions of the permutation w and Sn denotes the set of permutations of length n, we have
$\sum _{w\in S_{n}}q^{{\text{inv}}(w)}=[n]_{q}!.$
In particular, one recovers the usual factorial by taking the limit as $q\rightarrow 1$.
The q-factorial also has a concise definition in terms of the q-Pochhammer symbol, a basic building-block of all q-theories:
$[n]_{q}!={\frac {(q;q)_{n}}{(1-q)^{n}}}.$
From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:
${\binom {n}{k}}_{q}={\frac {[n]_{q}!}{[n-k]_{q}![k]_{q}!}}.$
The q-exponential is defined as:
$e_{q}(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{[n]_{q}!}}.$
q-trigonometric functions, along with a q-Fourier transform, have been defined in this context.
Combinatorial q-analogs
The Gaussian coefficients count subspaces of a finite vector space. Let q be the number of elements in a finite field. (The number q is then a power of a prime number, q = pe, so using the letter q is especially appropriate.) Then the number of k-dimensional subspaces of the n-dimensional vector space over the q-element field equals
${\binom {n}{k}}_{q}.$
Letting q approach 1, we get the binomial coefficient
${\binom {n}{k}},$
or in other words, the number of k-element subsets of an n-element set.
Thus, one can regard a finite vector space as a q-generalization of a set, and the subspaces as the q-generalization of the subsets of the set. This has been a fruitful point of view in finding interesting new theorems. For example, there are q-analogs of Sperner's theorem and Ramsey theory.
Cyclic sieving
Main article: Cyclic sieving
Let q = (e2πi/n)d be the d-th power of a primitive n-th root of unity. Let C be a cyclic group of order n generated by an element c. Let X be the set of k-element subsets of the n-element set {1, 2, ..., n}. The group C has a canonical action on X given by sending c to the cyclic permutation (1, 2, ..., n). Then the number of fixed points of cd on X is equal to
${\binom {n}{k}}_{q}.$
q → 1
Main article: Field with one element
Conversely, by letting q vary and seeing q-analogs as deformations, one can consider the combinatorial case of q = 1 as a limit of q-analogs as q → 1 (often one cannot simply let q = 1 in the formulae, hence the need to take a limit).
This can be formalized in the field with one element, which recovers combinatorics as linear algebra over the field with one element: for example, Weyl groups are simple algebraic groups over the field with one element.
Applications in the physical sciences
q-analogs are often found in exact solutions of many-body problems. In such cases, the q → 1 limit usually corresponds to relatively simple dynamics, e.g., without nonlinear interactions, while q < 1 gives insight into the complex nonlinear regime with feedbacks.
An example from atomic physics is the model of molecular condensate creation from an ultra cold fermionic atomic gas during a sweep of an external magnetic field through the Feshbach resonance.[3] This process is described by a model with a q-deformed version of the SU(2) algebra of operators, and its solution is described by q-deformed exponential and binomial distributions.
See also
• List of q-analogs
• Stirling number
• Young tableau
References
• Andrews, G. E., Askey, R. A. & Roy, R. (1999), Special Functions, Cambridge University Press, Cambridge.
• Gasper, G. & Rahman, M. (2004), Basic Hypergeometric Series, Cambridge University Press, ISBN 0521833574.
• Ismail, M. E. H. (2005), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press.
• Koekoek, R. & Swarttouw, R. F. (1998), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.
1. Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538
2. Ernst, Thomas (2003). "A Method for q-calculus" (PDF). Journal of Nonlinear Mathematical Physics. 10 (4): 487–525. Bibcode:2003JNMP...10..487E. doi:10.2991/jnmp.2003.10.4.5. Retrieved 2011-07-27.
3. C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution". Phys. Rev. A. 94 (3): 033808. arXiv:1606.08430. Bibcode:2016PhRvA..94c3808S. doi:10.1103/PhysRevA.94.033808.
External links
• "Umbral calculus", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• q-analog from MathWorld
• q-bracket from MathWorld
• q-factorial from MathWorld
• q-binomial coefficient from MathWorld
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Q-matrix
In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q.
This article is about the notion used in the context of linear complementarity problems. For the one used in the context of Markov-chain, see Continuous-time Markov chain § Definition.
Properties
• M is a Q-matrix if there exists d > 0 such that LCP(M,0) and LCP(M,d) have a unique solution.[1][2]
• Any P-matrix is a Q-matrix. Conversely, if a matrix is a Z-matrix and a Q-matrix, then it is also a P-matrix.[3]
See also
• P-matrix
• Z-matrix
References
1. Karamardian, S. (1976). "An existence theorem for the complementarity problem". Journal of Optimization Theory and Applications. 19 (2): 227–232. doi:10.1007/BF00934094. ISSN 0022-3239. S2CID 120505258.
2. Sivakumar, K. C.; Sushmitha, P.; Wendler, Megan (2020-05-17). "Karamardian Matrices: A Generalization of $Q$-Matrices". arXiv:2005.08171 [math.OC].
3. Berman, Abraham. (1994). Nonnegative matrices in the mathematical sciences. Plemmons, Robert J. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-321-8. OCLC 31206205.
• Murty, Katta G. (January 1972). "On the number of solutions to the complementarity problem and spanning properties of complementary cones" (PDF). Linear Algebra and Its Applications. 5 (1): 65–108. doi:10.1016/0024-3795(72)90019-5. hdl:2027.42/34188.
• Aganagic, Muhamed; Cottle, Richard W. (December 1979). "A note on Q-matrices". Mathematical Programming. 16 (1): 374–377. doi:10.1007/BF01582122. S2CID 6384105.
• Pang, Jong-Shi (December 1979). "On Q-matrices". Mathematical Programming. 17 (1): 243–247. doi:10.1007/BF01588247. S2CID 209858727.
• Danao, R. A. (November 1994). "Q-matrices and boundedness of solutions to linear complementarity problems". Journal of Optimization Theory and Applications. 83 (2): 321–332. doi:10.1007/bf02190060. S2CID 121165848.
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Q-systems
Q-systems are a method of directed graph transformations according to given grammar rules, developed at the Université de Montréal by Alain Colmerauer in 1967–70 for use in natural language processing. The Université de Montréal's machine translation system, TAUM-73, used the Q-Systems as its language formalism.
The data structure manipulated by a Q-system is a Q-graph, which is a directed acyclic graph with one entry node and one exit node, where each arc bears a labelled ordered tree. An input sentence is usually represented by a linear Q-graph where each arc bears a word (tree reduced to one node labelled by this word). After analysis, the Q-graph is usually a bundle of 1-arc paths, each arc bearing a possible analysis tree. After generation, the goal is usually to produce as many paths as desired outputs, with again one word per arc.
A Q-System consists of a sequence of Q-treatments, each being a set of Q-rules, of the form <matched_path> == <added_path> [<condition>]. The Q-treatments are applied in sequence, unless one of them produces the empty Q-graph, in which case the result is the last Q-graph obtained. The three parts of a rule can contain variables for labels, trees, and forests. All variables after "==" must appear in the <matched_path> part. Variables are local to rules.
A Q-treatment works in two steps, addition and cleaning. It first applies all its rules exhaustively, using instantiation (one-way unification), thereby adding new paths to the current Q-graph (added arcs and their trees can be used to produce new paths). If and when this addition process halts, all arcs used in some successful rule application are erased, as well as all unused arcs that are no more on any path from the entry node to the exit node. Hence, the result, if any (if the addition step terminates), is again a Q-graph. That allows several Q-Systems to be chained, each of them performing a specialized task, together forming a complex system. For example, TAUM 73 consisted of fifteen chained Q-Systems.
An extension of the basic idea of the Q-Systems, namely to replace instantiation by unification (to put it simply, allow "new" variables in the right hand side part of a rule, and replace parametrized labelled trees by logical terms) led to Prolog, designed by Alain Colmerauer and Philippe Roussel in 1972. Refinements in the other direction (reducing non-determinism and introducing typed labels) by John Chandioux led to GramR, used for programming METEO from 1985 onward.
In 2009, Hong Thai Nguyen of GETALP,[1] Laboratoire d'Informatique de Grenoble[2] reimplemented the Q-language in C, using ANTLR to compile the Q-systems and the Q-graphs, and an algorithm proposed by Christian Boitet (as none had been published and sources of the previous Fortran implementation had been lost). That implementation was corrected, completed and extended (to labels using Unicode characters and not only the printable characters of the CDC6600 of the historical version) by David Cattanéo in 2010-11.
See also
• METEO System
References
1. "Groupe d'Étude en Traduction Automatique/Traitement Automatisé des Langues et de la Parol" (in French).
2. "LIG" (in French).
Further reading
• Colmerauer, A: Les systèmes Q ou un formalisme pour analyser et synthétiser des phrases sur ordinateur. Mimeo, Montréal, 1969.
• Nguyen, H-T: Des systèmes de TA homogènes aux systèmes de TAO hétérogènes. thèse UJF, Grenoble, 2009.
External links
• http://unldeco.imag.fr/unldeco/SystemsQ.po?localhost=/home/nguyenht/SYS-Q/MONITEUR/] new Q-systems demonstration
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q-theta function
In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. [1][2] It is given by
$\theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)$
where one takes 0 ≤ |q| < 1. It obeys the identities
$\theta (z;q)=\theta \left({\frac {q}{z}};q\right)=-z\theta \left({\frac {1}{z}};q\right).$
It may also be expressed as:
$\theta (z;q)=(z;q)_{\infty }(q/z;q)_{\infty }$
where $(\cdot \cdot )_{\infty }$ is the q-Pochhammer symbol.
See also
• elliptic hypergeometric series
• Jacobi theta function
• Ramanujan theta function
References
1. Gasper, George; Rahman, Mizan (2004). Basic Hypergeometric Series. doi:10.1017/CBO9780511526251. ISBN 9780521833578.
2. Spiridonov, V. P. (2008). "Essays on the theory of elliptic hypergeometric functions". Russian Mathematical Surveys. 63 (3): 405–472. arXiv:0805.3135. Bibcode:2008RuMaS..63..405S. doi:10.1070/RM2008v063n03ABEH004533. S2CID 16996893.
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Bilinear quadrilateral element
The bilinear quadrilateral element, also known as the Q4 element,[1] is a type of element used in finite element analysis which is used to approximate in a 2D domain the exact solution to a given differential equation.
The element consists of a combination of two sets of Lagrange polynomials, each one used to define the variation of a field in each orthogonal direction of the local referential.
References
1. Zienkiewicz, O. C. (2000). The Finite Element Method Volume 1: The Basis (5th ed.).
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QAMA Calculator
The QAMA Calculator is a calculator that requires users to provide a reasonable estimate of the answer before the precise answer is delivered.[1][2] QAMA stands for Quick Approximate Mental Arithmetic.
Invented by Ilan Samson, it aims to get users to think first by estimating before they get the correct answer.[3] Estimation is seen by many as an essential part of mathematics, and some believe that the presence and popularity of calculators could inhibit the use of estimation skills.[4][5]
A physical version of the calculator was released for sale in 2014, with apps for smartphones and tablets developed in 2016.
References
1. "Introducing QAMA Calculator". qamacalculator.com.
2. Knapp, Alex. "QAMA: The Calculator That Makes You Better At Math".
3. Paul, Annie Murphy. "How to Use Technology to Make You Smarter". Time. ISSN 0040-781X. Retrieved 2018-08-23.
4. "A quote from What's Math Got to Do with It?". www.goodreads.com.
5. "Power of Estimation Takes Math Beyond Classroom - MU News Bureau". munews.missouri.edu.
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Wikipedia
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QARMA
QARMA (from Qualcomm ARM Authenticator[1]) is a lightweight tweakable block cipher primarily known for its use in the ARMv8 architecture for protection of software as a cryptographic hash for the Pointer Authentication Code.[2] The cipher was proposed by Roberto Avanzi in 2016.[2][3] Two versions of QARMA are defined: QARMA-64 (64-bit block size with a 128-bit encryption key) and QARMA-128 (128-bit block size with a 256-bit key). The design of the QARMA was influenced by PRINCE and MANTIS.[3] The cipher is intended for fully-unrolled hardware implementations with low latency (like memory encryption). Unlike the XTS mode, the address can be directly used as a tweak and does not need to be whitened with the block encryption first.
Architecture
QARMA is an Even-Mansour cipher using three stages, with whitening keys w0 and w1 XORed in between:
1. permutation F is using core key k0 and parameterized by a tweak T. It has r rounds inside (r = 7 for QARMA-64, r = 11 for QARMA-128);
2. "central" permutation C is using key k1 and is designed to be reversible via a simple key transformation (contains two central rounds);
3. the third permutation is an inverse of the first (r more rounds).
All keys are derived from the master encryption key K using specialisation:
• K is partitioned into halves as w0 Concatenation k0, each will have halfsize bits;
• for encryption w1 = (w0 >>> 1) + (w0 >> (halfsize-1));
• for encryption k1 = k0;
• for decryption, the same design can be used as long as k0+α is used as a core key, k1 = Q•k0, w1 and w0 are swapped. α here is a special constant and Q a special involutary matrix. This construct is similar to the alpha reflection in PRINCE.
The data is split into 16 cells (4-bit nibbles for QARMA-64, 8-bit bytes for QARMA-128). Internal state also contains 16 cells, arranged in a 4x4 matrix, and is initialized by plaintext (XORed with w0). In each round of $\digamma $, the state is transformed via operations $\tau ,M,S$:
• $\tau $ is ShuffleCells, a MIDORI permutation of cells ([ 0, 11, 6, 13, 10, 1, 12, 7, 5, 14, 3, 8, 15, 4, 9, 2]);
• $M$ is MixColumns: each column is multiplied by a fixed matrix M;
• $S$ is SubCells: each cell is transformed using an S-box.
The tweak for each round is updated using $h,\omega $:
• $h$ is a cell permutation from MANTIS ([ 6, 5, 14, 15, 0, 1, 2, 3, 7, 12, 13, 4, 8, 9, 10, 11]);
• $\omega $ is an LFSR applied to each of the cells with numbers [0, 1, 3, 4, 8, 11, 13]. For QARMA-64, the LFSR is (b3, b2, b1, b0) ⇒ (b0 + b1, b3, b2, b1), for QARMA-128, (b7, b6, ..., b0) ⇒ (b0 + b2, b7, b6, ..., b1),
The rounds of ${\overline {\digamma }}$ consist of inverse operations ${\overline {\tau }},{\overline {M}},{\overline {S}},{\overline {h}},{\overline {\omega }}$. Central rounds, in addition to two rounds ($\tau ,M,S$ and ${\overline {\tau }},{\overline {M}},{\overline {S}}$), include multiplication of the state by an involutary matrix Q.
References
1. Qameleon v. 1.0: A Submission to the NIST Lightweight Cryptography Standardizaࢼon Process
2. Zong & Dong 2016.
3. Avanzi 2016.
Sources
• Avanzi, Roberto (2016). The QARMA Block Cipher Family (PDF). IACR Transactions on Symmetric Cryptology (ToSC). Vol. 17 (published 8 March 2017). pp. 4–44. doi:10.13154/tosc.v2017.i1.4-44. Archived from the original (PDF) on May 13, 2020.
• Zong, Rui; Dong, Xiaoyang (2016). "Meet-in-the-Middle Attack on QARMA Block Cipher" (PDF). iacr.org. IACR. Retrieved 10 June 2022.
• Kaur, Jasmin; Kermani, Mehran Mozaffari; Azarderakhsh, Reza (1 January 2022). "Hardware Constructions for Lightweight Cryptographic Block Cipher QARMA With Error Detection Mechanisms". IEEE Transactions on Emerging Topics in Computing. 10 (1): 514–519. doi:10.1109/TETC.2020.3027789. eISSN 2376-4562. S2CID 226665710.
• Li, Rongjia; Jin, Chenhui (4 May 2018). "Meet-in-the-Middle Attacks on Reduced-Round QARMA-64/128". The Computer Journal. 61 (8): 1158–1165. doi:10.1093/comjnl/bxy045. eISSN 1460-2067. ISSN 0010-4620.
• Yang, Dong; Qi, Wen-feng; Chen, Hua-jin (2018). "Impossible Differential Attack on QARMA Family of Block Ciphers". Cryptology ePrint Archive.
External links
• Public-domain Python implementation of QARMA-64
• Open-source (MIT license) implementation of QARMA-64 in C
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Wikipedia
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Quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called color. Gluons are the force carriers of the theory, just as photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years.
Standard Model of particle physics
Elementary particles of the Standard Model
Background
Particle physics
Standard Model
Quantum field theory
Gauge theory
Spontaneous symmetry breaking
Higgs mechanism
Constituents
Electroweak interaction
Quantum chromodynamics
CKM matrix
Standard Model mathematics
Limitations
Strong CP problem
Hierarchy problem
Neutrino oscillations
Physics beyond the Standard Model
Scientists
• Rutherford
• Thomson
• Chadwick
• Bose
• Sudarshan
• Davis Jr
• Anderson
• Fermi
• Dirac
• Feynman
• Rubbia
• Gell-Mann
• Kendall
• Taylor
• Friedman
• Powell
• Anderson
• Glashow
• Iliopoulos
• Lederman
• Maiani
• Meer
• Cowan
• Nambu
• Chamberlain
• Cabibbo
• Schwartz
• Perl
• Majorana
• Weinberg
• Lee
• Ward
• Salam
• Kobayashi
• Maskawa
• Mills
• Yang
• Yukawa
• 't Hooft
• Veltman
• Gross
• Pais
• Pauli
• Politzer
• Reines
• Schwinger
• Wilczek
• Cronin
• Fitch
• Vleck
• Higgs
• Englert
• Brout
• Hagen
• Guralnik
• Kibble
• de Mayolo
• Lattes
• Zweig
QCD exhibits three salient properties:
• Color confinement. Due to the force between two color charges remaining constant as they are separated, the energy grows until a quark–antiquark pair is spontaneously produced, turning the initial hadron into a pair of hadrons instead of isolating a color charge. Although analytically unproven, color confinement is well established from lattice QCD calculations and decades of experiments.[1]
• Asymptotic freedom, a steady reduction in the strength of interactions between quarks and gluons as the energy scale of those interactions increases (and the corresponding length scale decreases). The asymptotic freedom of QCD was discovered in 1973 by David Gross and Frank Wilczek,[2] and independently by David Politzer in the same year.[3] For this work, all three shared the 2004 Nobel Prize in Physics.[4]
• Chiral symmetry breaking, the spontaneous symmetry breaking of an important global symmetry of quarks, detailed below, with the result of generating masses for hadrons far above the masses of the quarks, and making pseudoscalar mesons exceptionally light. Yoichiro Nambu was awarded the 2008 Nobel Prize in Physics for elucidating the phenomenon, a dozen years before the advent of QCD. Lattice simulations have confirmed all his generic predictions.
Terminology
Physicist Murray Gell-Mann coined the word quark in its present sense. It originally comes from the phrase "Three quarks for Muster Mark" in Finnegans Wake by James Joyce. On June 27, 1978, Gell-Mann wrote a private letter to the editor of the Oxford English Dictionary, in which he related that he had been influenced by Joyce's words: "The allusion to three quarks seemed perfect." (Originally, only three quarks had been discovered.)[5]
The three kinds of charge in QCD (as opposed to one in quantum electrodynamics or QED) are usually referred to as "color charge" by loose analogy to the three kinds of color (red, green and blue) perceived by humans. Other than this nomenclature, the quantum parameter "color" is completely unrelated to the everyday, familiar phenomenon of color.
The force between quarks is known as the colour force[6] (or color force[7]) or strong interaction, and is responsible for the nuclear force.
Since the theory of electric charge is dubbed "electrodynamics", the Greek word χρῶμα chroma "color" is applied to the theory of color charge, "chromodynamics".
History
With the invention of bubble chambers and spark chambers in the 1950s, experimental particle physics discovered a large and ever-growing number of particles called hadrons. It seemed that such a large number of particles could not all be fundamental. First, the particles were classified by charge and isospin by Eugene Wigner and Werner Heisenberg; then, in 1953–56,[8][9][10] according to strangeness by Murray Gell-Mann and Kazuhiko Nishijima (see Gell-Mann–Nishijima formula). To gain greater insight, the hadrons were sorted into groups having similar properties and masses using the eightfold way, invented in 1961 by Gell-Mann[11] and Yuval Ne'eman. Gell-Mann and George Zweig, correcting an earlier approach of Shoichi Sakata, went on to propose in 1963 that the structure of the groups could be explained by the existence of three flavors of smaller particles inside the hadrons: the quarks. Gell-Mann also briefly discussed a field theory model in which quarks interact with gluons.[12][13]
Perhaps the first remark that quarks should possess an additional quantum number was made[14] as a short footnote in the preprint of Boris Struminsky[15] in connection with the Ω− hyperon being composed of three strange quarks with parallel spins (this situation was peculiar, because since quarks are fermions, such a combination is forbidden by the Pauli exclusion principle):
Three identical quarks cannot form an antisymmetric S-state. In order to realize an antisymmetric orbital S-state, it is necessary for the quark to have an additional quantum number.
— B. V. Struminsky, Magnetic moments of barions in the quark model, JINR-Preprint P-1939, Dubna, Submitted on January 7, 1965
Boris Struminsky was a PhD student of Nikolay Bogolyubov. The problem considered in this preprint was suggested by Nikolay Bogolyubov, who advised Boris Struminsky in this research.[15] In the beginning of 1965, Nikolay Bogolyubov, Boris Struminsky and Albert Tavkhelidze wrote a preprint with a more detailed discussion of the additional quark quantum degree of freedom.[16] This work was also presented by Albert Tavkhelidze without obtaining consent of his collaborators for doing so at an international conference in Trieste (Italy), in May 1965.[17][18]
A similar mysterious situation was with the Δ++ baryon; in the quark model, it is composed of three up quarks with parallel spins. In 1964–65, Greenberg[19] and Han–Nambu[20] independently resolved the problem by proposing that quarks possess an additional SU(3) gauge degree of freedom, later called color charge. Han and Nambu noted that quarks might interact via an octet of vector gauge bosons: the gluons.
Since free quark searches consistently failed to turn up any evidence for the new particles, and because an elementary particle back then was defined as a particle that could be separated and isolated, Gell-Mann often said that quarks were merely convenient mathematical constructs, not real particles. The meaning of this statement was usually clear in context: He meant quarks are confined, but he also was implying that the strong interactions could probably not be fully described by quantum field theory.
Richard Feynman argued that high energy experiments showed quarks are real particles: he called them partons (since they were parts of hadrons). By particles, Feynman meant objects that travel along paths, elementary particles in a field theory.
The difference between Feynman's and Gell-Mann's approaches reflected a deep split in the theoretical physics community. Feynman thought the quarks have a distribution of position or momentum, like any other particle, and he (correctly) believed that the diffusion of parton momentum explained diffractive scattering. Although Gell-Mann believed that certain quark charges could be localized, he was open to the possibility that the quarks themselves could not be localized because space and time break down. This was the more radical approach of S-matrix theory.
James Bjorken proposed that pointlike partons would imply certain relations in deep inelastic scattering of electrons and protons, which were verified in experiments at SLAC in 1969. This led physicists to abandon the S-matrix approach for the strong interactions.
In 1973 the concept of color as the source of a "strong field" was developed into the theory of QCD by physicists Harald Fritzsch and Heinrich Leutwyler, together with physicist Murray Gell-Mann.[21] In particular, they employed the general field theory developed in 1954 by Chen Ning Yang and Robert Mills[22] (see Yang–Mills theory), in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.)
The discovery of asymptotic freedom in the strong interactions by David Gross, David Politzer and Frank Wilczek allowed physicists to make precise predictions of the results of many high energy experiments using the quantum field theory technique of perturbation theory. Evidence of gluons was discovered in three-jet events at PETRA in 1979. These experiments became more and more precise, culminating in the verification of perturbative QCD at the level of a few percent at LEP, at CERN.
The other side of asymptotic freedom is confinement. Since the force between color charges does not decrease with distance, it is believed that quarks and gluons can never be liberated from hadrons. This aspect of the theory is verified within lattice QCD computations, but is not mathematically proven. One of the Millennium Prize Problems announced by the Clay Mathematics Institute requires a claimant to produce such a proof. Other aspects of non-perturbative QCD are the exploration of phases of quark matter, including the quark–gluon plasma.
The relation between the short-distance particle limit and the confining long-distance limit is one of the topics recently explored using string theory, the modern form of S-matrix theory.[23][24]
Theory
Some definitions
Unsolved problem in physics:
QCD in the non-perturbative regime:
• Confinement: the equations of QCD remain unsolved at energy scales relevant for describing atomic nuclei. How does QCD give rise to the physics of nuclei and nuclear constituents?
• Quark matter: the equations of QCD predict that a plasma (or soup) of quarks and gluons should be formed at high temperature and density. What are the properties of this phase of matter?
(more unsolved problems in physics)
Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations. These can be
• local symmetries, which are the symmetries that act independently at each point in spacetime. Each such symmetry is the basis of a gauge theory and requires the introduction of its own gauge bosons.
• global symmetries, which are symmetries whose operations must be simultaneously applied to all points of spacetime.
QCD is a non-abelian gauge theory (or Yang–Mills theory) of the SU(3) gauge group obtained by taking the color charge to define a local symmetry.
Since the strong interaction does not discriminate between different flavors of quark, QCD has approximate flavor symmetry, which is broken by the differing masses of the quarks.
There are additional global symmetries whose definitions require the notion of chirality, discrimination between left and right-handed. If the spin of a particle has a positive projection on its direction of motion then it is called right-handed; otherwise, it is left-handed. Chirality and handedness are not the same, but become approximately equivalent at high energies.
• Chiral symmetries involve independent transformations of these two types of particle.
• Vector symmetries (also called diagonal symmetries) mean the same transformation is applied on the two chiralities.
• Axial symmetries are those in which one transformation is applied on left-handed particles and the inverse on the right-handed particles.
Additional remarks: duality
As mentioned, asymptotic freedom means that at large energy – this corresponds also to short distances – there is practically no interaction between the particles. This is in contrast – more precisely one would say dual– to what one is used to, since usually one connects the absence of interactions with large distances. However, as already mentioned in the original paper of Franz Wegner,[25] a solid state theorist who introduced 1971 simple gauge invariant lattice models, the high-temperature behaviour of the original model, e.g. the strong decay of correlations at large distances, corresponds to the low-temperature behaviour of the (usually ordered!) dual model, namely the asymptotic decay of non-trivial correlations, e.g. short-range deviations from almost perfect arrangements, for short distances. Here, in contrast to Wegner, we have only the dual model, which is that one described in this article.[26]
Symmetry groups
The color group SU(3) corresponds to the local symmetry whose gauging gives rise to QCD. The electric charge labels a representation of the local symmetry group U(1), which is gauged to give QED: this is an abelian group. If one considers a version of QCD with Nf flavors of massless quarks, then there is a global (chiral) flavor symmetry group SUL(Nf) × SUR(Nf) × UB(1) × UA(1). The chiral symmetry is spontaneously broken by the QCD vacuum to the vector (L+R) SUV(Nf) with the formation of a chiral condensate. The vector symmetry, UB(1) corresponds to the baryon number of quarks and is an exact symmetry. The axial symmetry UA(1) is exact in the classical theory, but broken in the quantum theory, an occurrence called an anomaly. Gluon field configurations called instantons are closely related to this anomaly.
There are two different types of SU(3) symmetry: there is the symmetry that acts on the different colors of quarks, and this is an exact gauge symmetry mediated by the gluons, and there is also a flavor symmetry that rotates different flavors of quarks to each other, or flavor SU(3). Flavor SU(3) is an approximate symmetry of the vacuum of QCD, and is not a fundamental symmetry at all. It is an accidental consequence of the small mass of the three lightest quarks.
In the QCD vacuum there are vacuum condensates of all the quarks whose mass is less than the QCD scale. This includes the up and down quarks, and to a lesser extent the strange quark, but not any of the others. The vacuum is symmetric under SU(2) isospin rotations of up and down, and to a lesser extent under rotations of up, down, and strange, or full flavor group SU(3), and the observed particles make isospin and SU(3) multiplets.
The approximate flavor symmetries do have associated gauge bosons, observed particles like the rho and the omega, but these particles are nothing like the gluons and they are not massless. They are emergent gauge bosons in an approximate string description of QCD.
Lagrangian
The dynamics of the quarks and gluons are controlled by the quantum chromodynamics Lagrangian. The gauge invariant QCD Lagrangian is
${\mathcal {L}}_{\mathrm {QCD} }={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }$
where $\psi _{i}(x)\,$ is the quark field, a dynamical function of spacetime, in the fundamental representation of the SU(3) gauge group, indexed by $i$ and $j$ running from $1$ to $3$; $D_{\mu }$ is the gauge covariant derivative; the γμ are Gamma matrices connecting the spinor representation to the vector representation of the Lorentz group.
Herein, the gauge covariant derivative $\left(D_{\mu }\right)_{ij}=\partial _{\mu }\delta _{ij}-ig\left(T_{a}\right)_{ij}{\mathcal {A}}_{\mu }^{a}\,$couples the quark field with a coupling strength $g\,$to the gluon fields via the infinitesimal SU(3) generators $T_{a}\,$in the fundamental representation. An explicit representation of these generators is given by $T_{a}=\lambda _{a}/2\,$, wherein the $\lambda _{a}\,(a=1\ldots 8)\,$are the Gell-Mann matrices.
The symbol $G_{\mu \nu }^{a}\,$ represents the gauge invariant gluon field strength tensor, analogous to the electromagnetic field strength tensor, Fμν, in quantum electrodynamics. It is given by:[27]
$G_{\mu \nu }^{a}=\partial _{\mu }{\mathcal {A}}_{\nu }^{a}-\partial _{\nu }{\mathcal {A}}_{\mu }^{a}+gf^{abc}{\mathcal {A}}_{\mu }^{b}{\mathcal {A}}_{\nu }^{c}\,,$
where ${\mathcal {A}}_{\mu }^{a}(x)\,$ are the gluon fields, dynamical functions of spacetime, in the adjoint representation of the SU(3) gauge group, indexed by a, b and c running from $1$ to $8$; and fabc are the structure constants of SU(3) (the generators of the adjoint representation). Note that the rules to move-up or pull-down the a, b, or c indices are trivial, (+, ..., +), so that fabc = fabc = fabc whereas for the μ or ν indices one has the non-trivial relativistic rules corresponding to the metric signature (+ − − −).
The variables m and g correspond to the quark mass and coupling of the theory, respectively, which are subject to renormalization.
An important theoretical concept is the Wilson loop (named after Kenneth G. Wilson). In lattice QCD, the final term of the above Lagrangian is discretized via Wilson loops, and more generally the behavior of Wilson loops can distinguish confined and deconfined phases.
Fields
Quarks are massive spin-1⁄2 fermions that carry a color charge whose gauging is the content of QCD. Quarks are represented by Dirac fields in the fundamental representation 3 of the gauge group SU(3). They also carry electric charge (either −1⁄3 or +2⁄3) and participate in weak interactions as part of weak isospin doublets. They carry global quantum numbers including the baryon number, which is 1⁄3 for each quark, hypercharge and one of the flavor quantum numbers.
Gluons are spin-1 bosons that also carry color charges, since they lie in the adjoint representation 8 of SU(3). They have no electric charge, do not participate in the weak interactions, and have no flavor. They lie in the singlet representation 1 of all these symmetry groups.
Each type of quark has a corresponding antiquark, of which the charge is exactly opposite. They transform in the conjugate representation to quarks, denoted ${\bar {\mathbf {3} }}$.
Dynamics
According to the rules of quantum field theory, and the associated Feynman diagrams, the above theory gives rise to three basic interactions: a quark may emit (or absorb) a gluon, a gluon may emit (or absorb) a gluon, and two gluons may directly interact. This contrasts with QED, in which only the first kind of interaction occurs, since photons have no charge. Diagrams involving Faddeev–Popov ghosts must be considered too (except in the unitarity gauge).
Area law and confinement
Detailed computations with the above-mentioned Lagrangian[28] show that the effective potential between a quark and its anti-quark in a meson contains a term that increases in proportion to the distance between the quark and anti-quark ($\propto r$), which represents some kind of "stiffness" of the interaction between the particle and its anti-particle at large distances, similar to the entropic elasticity of a rubber band (see below). This leads to confinement [29] of the quarks to the interior of hadrons, i.e. mesons and nucleons, with typical radii Rc, corresponding to former "Bag models" of the hadrons[30] The order of magnitude of the "bag radius" is 1 fm (= 10−15 m). Moreover, the above-mentioned stiffness is quantitatively related to the so-called "area law" behavior of the expectation value of the Wilson loop product PW of the ordered coupling constants around a closed loop W; i.e. $\,\langle P_{W}\rangle $ is proportional to the area enclosed by the loop. For this behavior the non-abelian behavior of the gauge group is essential.
Methods
Further analysis of the content of the theory is complicated. Various techniques have been developed to work with QCD. Some of them are discussed briefly below.
Perturbative QCD
This approach is based on asymptotic freedom, which allows perturbation theory to be used accurately in experiments performed at very high energies. Although limited in scope, this approach has resulted in the most precise tests of QCD to date.
Lattice QCD
Among non-perturbative approaches to QCD, the most well established is lattice QCD. This approach uses a discrete set of spacetime points (called the lattice) to reduce the analytically intractable path integrals of the continuum theory to a very difficult numerical computation that is then carried out on supercomputers like the QCDOC, which was constructed for precisely this purpose. While it is a slow and resource-intensive approach, it has wide applicability, giving insight into parts of the theory inaccessible by other means, in particular into the explicit forces acting between quarks and antiquarks in a meson. However, the numerical sign problem makes it difficult to use lattice methods to study QCD at high density and low temperature (e.g. nuclear matter or the interior of neutron stars).
1/N expansion
A well-known approximation scheme, the 1⁄N expansion, starts from the idea that the number of colors is infinite, and makes a series of corrections to account for the fact that it is not. Until now, it has been the source of qualitative insight rather than a method for quantitative predictions. Modern variants include the AdS/CFT approach.
Effective theories
For specific problems, effective theories may be written down that give qualitatively correct results in certain limits. In the best of cases, these may then be obtained as systematic expansions in some parameters of the QCD Lagrangian. One such effective field theory is chiral perturbation theory or ChiPT, which is the QCD effective theory at low energies. More precisely, it is a low energy expansion based on the spontaneous chiral symmetry breaking of QCD, which is an exact symmetry when quark masses are equal to zero, but for the u, d and s quark, which have small mass, it is still a good approximate symmetry. Depending on the number of quarks that are treated as light, one uses either SU(2) ChiPT or SU(3) ChiPT. Other effective theories are heavy quark effective theory (which expands around heavy quark mass near infinity), and soft-collinear effective theory (which expands around large ratios of energy scales). In addition to effective theories, models like the Nambu–Jona-Lasinio model and the chiral model are often used when discussing general features.
QCD sum rules
Based on an Operator product expansion one can derive sets of relations that connect different observables with each other.
Experimental tests
The notion of quark flavors was prompted by the necessity of explaining the properties of hadrons during the development of the quark model. The notion of color was necessitated by the puzzle of the
Δ++
. This has been dealt with in the section on the history of QCD.
The first evidence for quarks as real constituent elements of hadrons was obtained in deep inelastic scattering experiments at SLAC. The first evidence for gluons came in three-jet events at PETRA.[32]
Several good quantitative tests of perturbative QCD exist:
• The running of the QCD coupling as deduced from many observations
• Scaling violation in polarized and unpolarized deep inelastic scattering
• Vector boson production at colliders (this includes the Drell–Yan process)
• Direct photons produced in hadronic collisions
• Jet cross sections in colliders
• Event shape observables at the LEP
• Heavy-quark production in colliders
Quantitative tests of non-perturbative QCD are fewer, because the predictions are harder to make. The best is probably the running of the QCD coupling as probed through lattice computations of heavy-quarkonium spectra. There is a recent claim about the mass of the heavy meson Bc . Other non-perturbative tests are currently at the level of 5% at best. Continuing work on masses and form factors of hadrons and their weak matrix elements are promising candidates for future quantitative tests. The whole subject of quark matter and the quark–gluon plasma is a non-perturbative test bed for QCD that still remains to be properly exploited.
One qualitative prediction of QCD is that there exist composite particles made solely of gluons called glueballs that have not yet been definitively observed experimentally. A definitive observation of a glueball with the properties predicted by QCD would strongly confirm the theory. In principle, if glueballs could be definitively ruled out, this would be a serious experimental blow to QCD. But, as of 2013, scientists are unable to confirm or deny the existence of glueballs definitively, despite the fact that particle accelerators have sufficient energy to generate them.
Cross-relations to condensed matter physics
There are unexpected cross-relations to condensed matter physics. For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses,[33] which are systems with the usual spin degrees of freedom $s_{i}=\pm 1\,$ for i =1,...,N, with the special fixed "random" couplings $J_{i,k}=\epsilon _{i}\,J_{0}\,\epsilon _{k}\,.$ Here the εi and εk quantities can independently and "randomly" take the values ±1, which corresponds to a most-simple gauge transformation $(\,s_{i}\to s_{i}\cdot \epsilon _{i}\quad \,J_{i,k}\to \epsilon _{i}J_{i,k}\epsilon _{k}\,\quad s_{k}\to s_{k}\cdot \epsilon _{k}\,)\,.$ This means that thermodynamic expectation values of measurable quantities, e.g. of the energy $ {\mathcal {H}}:=-\sum s_{i}\,J_{i,k}\,s_{k}\,,$ are invariant.
However, here the coupling degrees of freedom $J_{i,k}$, which in the QCD correspond to the gluons, are "frozen" to fixed values (quenching). In contrast, in the QCD they "fluctuate" (annealing), and through the large number of gauge degrees of freedom the entropy plays an important role (see below).
For positive J0 the thermodynamics of the Mattis spin glass corresponds in fact simply to a "ferromagnet in disguise", just because these systems have no "frustration" at all. This term is a basic measure in spin glass theory.[34] Quantitatively it is identical with the loop product $P_{W}:\,=\,J_{i,k}J_{k,l}...J_{n,m}J_{m,i}$ along a closed loop W. However, for a Mattis spin glass – in contrast to "genuine" spin glasses – the quantity PW never becomes negative.
The basic notion "frustration" of the spin-glass is actually similar to the Wilson loop quantity of the QCD. The only difference is again that in the QCD one is dealing with SU(3) matrices, and that one is dealing with a "fluctuating" quantity. Energetically, perfect absence of frustration should be non-favorable and atypical for a spin glass, which means that one should add the loop product to the Hamiltonian, by some kind of term representing a "punishment". In the QCD the Wilson loop is essential for the Lagrangian rightaway.
The relation between the QCD and "disordered magnetic systems" (the spin glasses belong to them) were additionally stressed in a paper by Fradkin, Huberman and Shenker,[35] which also stresses the notion of duality.
A further analogy consists in the already mentioned similarity to polymer physics, where, analogously to Wilson loops, so-called "entangled nets" appear, which are important for the formation of the entropy-elasticity (force proportional to the length) of a rubber band. The non-abelian character of the SU(3) corresponds thereby to the non-trivial "chemical links", which glue different loop segments together, and "asymptotic freedom" means in the polymer analogy simply the fact that in the short-wave limit, i.e. for $0\leftarrow \lambda _{w}\ll R_{c}$ (where Rc is a characteristic correlation length for the glued loops, corresponding to the above-mentioned "bag radius", while λw is the wavelength of an excitation) any non-trivial correlation vanishes totally, as if the system had crystallized.[36]
There is also a correspondence between confinement in QCD – the fact that the color field is only different from zero in the interior of hadrons – and the behaviour of the usual magnetic field in the theory of type-II superconductors: there the magnetism is confined to the interior of the Abrikosov flux-line lattice,[37] i.e., the London penetration depth λ of that theory is analogous to the confinement radius Rc of quantum chromodynamics. Mathematically, this correspondendence is supported by the second term, $\propto gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}\,,$ on the r.h.s. of the Lagrangian.
See also
• For overviews:
• Standard Model
• Strong interaction
• Quark
• Gluon
• Hadron
• Color confinement
• QCD matter
• Quark–gluon plasma
• For details:
• Gauge theory
• Quantum gauge theory, BRST quantization and Faddeev–Popov ghost
• Quantum field theory – a more general category
• For techniques:
• Lattice QCD
• 1/N expansion
• Perturbative QCD
• Soft-collinear effective theory
• Heavy quark effective theory
• Chiral model
• Nambu–Jona-Lasinio model
• For experiments:
• Deep inelastic scattering
• Jet (particle physics)
• Quark–gluon plasma
• Quantum electrodynamics
• Symmetry in quantum mechanics
• Yang–Mills theory
• Yang–Mills existence and mass gap
References
1. J. Greensite (2011). An introduction to the confinement problem. Springer. ISBN 978-3-642-14381-6.
2. D.J. Gross; F. Wilczek (1973). "Ultraviolet behavior of non-abelian gauge theories". Physical Review Letters. 30 (26): 1343–1346. Bibcode:1973PhRvL..30.1343G. doi:10.1103/PhysRevLett.30.1343.
3. H.D. Politzer (1973). "Reliable perturbative results for strong interactions". Physical Review Letters. 30 (26): 1346–1349. Bibcode:1973PhRvL..30.1346P. doi:10.1103/PhysRevLett.30.1346.
4. "The Nobel Prize in Physics 2004". Nobel Web. 2004. Archived from the original on 2010-11-06. Retrieved 2010-10-24.
5. Gell-Mann, Murray (1995). The Quark and the Jaguar. Owl Books. ISBN 978-0-8050-7253-2.
6. wikt:colour force
7. "The Color Force". Archived from the original on 2007-08-20. Retrieved 2007-08-29. retrieved 6 May 2017
8. Nakano, T; Nishijima, N (1953). "Charge Independence for V-particles". Progress of Theoretical Physics. 10 (5): 581. Bibcode:1953PThPh..10..581N. doi:10.1143/PTP.10.581.
9. Nishijima, K (1955). "Charge Independence Theory of V Particles". Progress of Theoretical Physics. 13 (3): 285–304. Bibcode:1955PThPh..13..285N. doi:10.1143/PTP.13.285.
10. Gell-Mann, M (1956). "The Interpretation of the New Particles as Displaced Charged Multiplets". Il Nuovo Cimento. 4 (S2): 848–866. Bibcode:1956NCim....4S.848G. doi:10.1007/BF02748000. S2CID 121017243.
11. Gell-Mann, M. (1961). "The Eightfold Way: A Theory of strong interaction symmetry" (No. TID-12608; CTSL-20). California Inst. of Tech., Pasadena. Synchrotron Lab (online).
12. M. Gell-Mann (1964). "A Schematic Model of Baryons and Mesons". Physics Letters. 8 (3): 214–215. Bibcode:1964PhL.....8..214G. doi:10.1016/S0031-9163(64)92001-3.
13. M. Gell-Mann; H. Fritzsch (2010). Murray Gell-Mann: Selected Papers. World Scientific. Bibcode:2010mgsp.book.....F.
14. Fyodor Tkachov (2009). "A contribution to the history of quarks: Boris Struminsky's 1965 JINR publication". arXiv:0904.0343 [physics.hist-ph].
15. B. V. Struminsky, Magnetic moments of baryons in the quark model. JINR-Preprint P-1939, Dubna, Russia. Submitted on January 7, 1965.
16. N. Bogolubov, B. Struminsky, A. Tavkhelidze. On composite models in the theory of elementary particles. JINR Preprint D-1968, Dubna 1965.
17. A. Tavkhelidze. Proc. Seminar on High Energy Physics and Elementary Particles, Trieste, 1965, Vienna IAEA, 1965, p. 763.
18. V. A. Matveev and A. N. Tavkhelidze (INR, RAS, Moscow) The quantum number color, colored quarks and QCD Archived 2007-05-23 at the Wayback Machine (Dedicated to the 40th Anniversary of the Discovery of the Quantum Number Color). Report presented at the 99th Session of the JINR Scientific Council, Dubna, 19–20 January 2006.
19. Greenberg, O. W. (1964). "Spin and Unitary Spin Independence in a Paraquark Model of Baryons and Mesons". Phys. Rev. Lett. 13 (20): 598–602. Bibcode:1964PhRvL..13..598G. doi:10.1103/PhysRevLett.13.598.
20. Han, M. Y.; Nambu, Y. (1965). "Three-Triplet Model with Double SU(3) Symmetry". Phys. Rev. 139 (4B): B1006–B1010. Bibcode:1965PhRv..139.1006H. doi:10.1103/PhysRev.139.B1006.
21. Fritzsch, H.; Gell-Mann, M.; Leutwyler, H. (1973). "Advantages of the color octet gluon picture". Physics Letters. 47B (4): 365–368. Bibcode:1973PhLB...47..365F. CiteSeerX 10.1.1.453.4712. doi:10.1016/0370-2693(73)90625-4.
22. Yang, C. N.; Mills, R. (1954). "Conservation of Isotopic Spin and Isotopic Gauge Invariance". Physical Review. 96 (1): 191–195. Bibcode:1954PhRv...96..191Y. doi:10.1103/PhysRev.96.191.
23. J. Polchinski; M. Strassler (2002). "Hard Scattering and Gauge/String duality". Physical Review Letters. 88 (3): 31601. arXiv:hep-th/0109174. Bibcode:2002PhRvL..88c1601P. doi:10.1103/PhysRevLett.88.031601. PMID 11801052. S2CID 2891297.
24. Brower, Richard C.; Mathur, Samir D.; Chung-I Tan (2000). "Glueball Spectrum for QCD from AdS Supergravity Duality". Nuclear Physics B. 587 (1–3): 249–276. arXiv:hep-th/0003115. Bibcode:2000NuPhB.587..249B. doi:10.1016/S0550-3213(00)00435-1. S2CID 11971945.
25. Wegner, F. (1971). "Duality in Generalized Ising Models and Phase Transitions without Local Order Parameter". J. Math. Phys. 12 (10): 2259–2272. Bibcode:1971JMP....12.2259W. doi:10.1063/1.1665530. Reprinted in Rebbi, Claudio, ed. (1983). Lattice Gauge Theories and Monte Carlo Simulations. Singapore: World Scientific. pp. 60–73. ISBN 9971950707. Abstract: Archived 2011-05-04 at the Wayback Machine
26. Perhaps one can guess that in the "original" model mainly the quarks would fluctuate, whereas in the present one, the "dual" model, mainly the gluons do.
27. M. Eidemüller; H.G. Dosch; M. Jamin (2000). "The field strength correlator from QCD sum rules". Nucl. Phys. B Proc. Suppl. Heidelberg, Germany. 86 (1–3): 421–425. arXiv:hep-ph/9908318. Bibcode:2000NuPhS..86..421E. doi:10.1016/S0920-5632(00)00598-3. S2CID 18237543.
28. See all standard textbooks on the QCD, e.g., those noted above
29. Confinement gives way to a quark–gluon plasma only at extremely large pressures and/or temperatures, e.g. for $T\approx 5\cdot 10^{12}$ K or larger.
30. Kenneth Alan Johnson. (July 1979). The bag model of quark confinement. Scientific American.
31. Cardoso, M.; et al. (2010). "Lattice QCD computation of the colour fields for the static hybrid quark–gluon–antiquark system, and microscopic study of the Casimir scaling". Phys. Rev. D. 81 (3): 034504. arXiv:0912.3181. Bibcode:2010PhRvD..81c4504C. doi:10.1103/PhysRevD.81.034504. S2CID 119216789.
32. Bethke, S. (2007-04-01). "Experimental tests of asymptotic freedom". Progress in Particle and Nuclear Physics. 58 (2): 351–386. arXiv:hep-ex/0606035. Bibcode:2007PrPNP..58..351B. doi:10.1016/j.ppnp.2006.06.001. ISSN 0146-6410. S2CID 14915298.
33. Mattis, D. C. (1976). "Solvable Spin Systems with Random Interactions". Phys. Lett. A. 56 (5): 421–422. Bibcode:1976PhLA...56..421M. doi:10.1016/0375-9601(76)90396-0.
34. Vannimenus, J.; Toulouse, G. (1977). "Theory of the frustration effect. II. Ising spins on a square lattice". Journal of Physics C: Solid State Physics. 10 (18): 537. Bibcode:1977JPhC...10L.537V. doi:10.1088/0022-3719/10/18/008.
35. Fradkin, Eduardo (1978). "Gauge symmetries in random magnetic systems" (PDF). Physical Review B. 18 (9): 4789–4814. Bibcode:1978PhRvB..18.4789F. doi:10.1103/physrevb.18.4789. OSTI 1446867.
36. Bergmann, A.; Owen, A. (2004). "Dielectric relaxation spectroscopy of poly[(R)-3-Hydroxybutyrate] (PHD) during crystallization". Polymer International. 53 (7): 863–868. doi:10.1002/pi.1445.
37. Mathematically, the flux-line lattices are described by Emil Artin's braid group, which is nonabelian, since one braid can wind around another one.
Further reading
• Greiner, Walter; Schramm, Stefan; Stein, Eckart (2007). Quantum Chromodynamics. Berlin Heidelberg: Springer. ISBN 978-3-540-48535-3.
• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 978-0-471-88741-6.
• Creutz, Michael (1985). Quarks, Gluons and Lattices. Cambridge University Press. ISBN 978-0-521-31535-7.
External links
• Frank Wilczek (2000). "QCD made simple" (PDF). Physics Today. 53 (8): 22–28. Bibcode:2000PhT....53h..22W. doi:10.1063/1.1310117.
• Particle data group
• The millennium prize for proving confinement
• Ab Initio Determination of Light Hadron Masses
• Andreas S Kronfeld The Weight of the World Is Quantum Chromodynamics
• Andreas S Kronfeld Quantum chromodynamics with advanced computing
• Standard model gets right answer
• Quantum Chromodynamics
• Cern Courier, The history of QCD with Prof. Dr. Harald Fritzsch
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See also: Template:Quantum mechanics topics
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Wikipedia
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QED manifesto
The QED manifesto was a proposal for a computer-based database of all mathematical knowledge, strictly formalized and with all proofs having been checked automatically. (Q.E.D. means quod erat demonstrandum in Latin, meaning "which was to be demonstrated.")
Overview
The idea for the project arose in 1993, mainly under the impetus of Robert Boyer. The goals of the project, tentatively named QED project or project QED, were outlined in the QED manifesto, a document first published in 1994, with input from several researchers.[1] Explicit authorship was deliberately avoided. A dedicated mailing list was created, and two scientific conferences on QED took place, the first one in 1994 at Argonne National Laboratories and the second in 1995 in Warsaw organized by the Mizar group.[2]
The project seems to have dissolved by 1996, never having produced more than discussions and plans. In a 2007 paper, Freek Wiedijk identifies two reasons for the failure of the project.[3] In order of importance:
• Very few people are working on formalization of mathematics. There is no compelling application for fully mechanized mathematics.
• Formalized mathematics does not yet resemble real, traditional mathematics. This is partly due to the complexity of mathematical notation, and partly to the limitations of existing theorem provers and proof assistants; the paper finds that the major contenders, Mizar, HOL, and Coq, have serious shortcomings in their abilities to express mathematics.
Nonetheless, QED-style projects are regularly proposed. The Mizar Mathematical Library formalizes a large portion of undergraduate mathematics, and was considered the largest such library in 2007.[4] Similar projects include the Metamath proof database and the mathlib library written in Lean.[5]
In 2014 the Twenty years of the QED Manifesto[6] workshop was organized as part of the Vienna Summer of Logic.
See also
• Formalism (mathematics)
• Mathematical knowledge management
• POPLmark, a more modest project in programming language theory
References
1. The QED Manifesto in Automated Deduction - CADE 12, Springer-Verlag, Lecture Notes in Artificial Intelligence, Vol. 814, pp. 238-251, 1994. HTML version
2. The QED Workshop II report
3. Freek Wiedijk, The QED Manifesto Revisited, 2007
4. Fairouz Kamareddine, Manuel Maarek, Krzysztof Retel, and J. B. Wells, Gradual Computerisation/Formalisation of Mathematical Texts into Mizar
5. mathlib libraryhttps://leanprover-community.github.io/mathlib-overview.html
6. Twenty years of the QED Manifesto workshop
Further reading
• H. Barendregt & F. Wiedijk, The Challenge of Computer Mathematics, Transactions A of the Royal Society 363 no. 1835, 2351–2375, 2005
• "A Special Issue on Formal Proof". Notices of the American Mathematical Society. December 2008. (open access issue)
• Richard A. De Millo, Richard J. Lipton, Alan J. Perlis, Social processes and proofs of theorems and programs, Communications of the ACM, Volume 22, Issue 5 (May 1979), Pages: 271 - 280
• John Harrison, Formalized Mathematics, Technical Report 36, Turku Centre for Computer Science (TUCS)
• Ittay Weiss, The QED Manifesto after Two Decades Version 2.0, Journal of Software vol. 11, no. 8, pp. 803-815, 2016.
External links
• Freek Wiedijk, Formalizing 100 Theorems A page keeping track of the progress in the formalization of 100 common theorems.
• Freek Wiedijk, The Seventeen Provers of the World, a proof of the irrationality of the square root of two in seventeen different proof assistants.
• Formalized Mathematics a journal in which Mizar proofs are presented.
• The Archive of Formal Proofs a similar (refereed) repository of proofs in Isabelle/HOL.
• A repository of proofs in Coq.
• UniMath "Coq library aims to formalize a substantial body of mathematics using the univalent point of view"
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Wikipedia
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QLattice
The QLattice is a software library which provides a framework for symbolic regression in Python. It works on Linux, Windows, and macOS. The QLattice algorithm is developed by the Danish/Spanish AI research company Abzu.[1] Since its creation, the QLattice has attracted significant attention, mainly for the inherent explainability of the models it produces.[2][3][4]
QLattice
Developer(s)Abzu
Initial releaseMarch 4, 2020 (2020-03-04)
Written inC, Python
Operating systemLinux, macOS, Windows
TypeMachine learning
LicenseCC BY-NC-ND 4.0
Websitedocs.abzu.ai
At the GECCO conference in Boston, MA in July 2022, the QLattice was announced as the winner of the synthetic track of the SRBench competition.[5]
Features
The QLattice works with data in categorical and numeric format. It allows the user to quickly generate, plot and inspect mathematical formulae that can potentially explain the generating process of the data. It is designed for easy interaction with the researcher, allowing the user to guide the search based on their preexisting knowledge.[2][6]
Scientific results
The QLattice mainly targets scientists, and integrates well with the scientific workflow.[2][6] It has been used in research into many different areas, such as energy consumption in buildings,[3] water potability,[7] heart failure,[8] pre-eclampsia,[4] Alzheimer’s disease,[9] hepatocellular carcinoma,[9] and breast cancer.[9]
See also
• Symbolic regression
• Explainable artificial intelligence
References
1. Kevin René Broløs; Meera Vieira Machado; Chris Cave; Jaan Kasak; Valdemar Stentoft-Hansen; Victor Galindo Batanero; Tom Jelen; Casper Wilstrup (2021-04-12). "An Approach to Symbolic Regression Using Feyn". arXiv:2104.05417 [cs.LG].
2. Abzu (2022-07-22). "What is a QLattice?".
3. Wenninger, Simon; Kaymakci, Can; Wiethe, Christian (2022). "Explainable long-term building energy consumption prediction using QLattice". Applied Energy. Elsevier BV. 308: 118300. doi:10.1016/j.apenergy.2021.118300. ISSN 0306-2619. S2CID 245428233.
4. Wilstrup, Casper; Hedley, Paula L.; Rode, Line; Placing, Sophie; Wøjdemann, Karen R.; Shalmi, Anne-Cathrine; Sundberg, Karin; Christiansen, Michael (2022-06-30), Symbolic regression analysis of interactions between first trimester maternal serum adipokines in pregnancies which develop pre-eclampsia, Cold Spring Harbor Laboratory, doi:10.1101/2022.06.29.22277072, S2CID 250331945
5. Michael Kommenda; William La Cava; Maimuna Majumder; Fabricio Olivetti de França; Marco Virgolin (2022-07-22). "SRBench Competition 2022: Interpretable Symbolic Regression for Data Science".
6. Bharadi, Vinayak (2021-07-30). "QLattice Environment and Feyn QGraph Models—A New Perspective Toward Deep Learning". Emerging Technologies for Healthcare. Wiley. pp. 69–92. doi:10.1002/9781119792345.ch3. ISBN 9781119792345. S2CID 238793347.
7. Riyantoko, Prismahardi Aji; Diyasa, I Gede Susrama Mas (2021-10-28). "F.Q.A.M" Feyn-QLattice Automation Modelling: Python Module of Machine Learning for Data Classification in Water Potability. IEEE. pp. 135–141. doi:10.1109/icimcis53775.2021.9699371. ISBN 978-1-6654-2733-3.
8. Wilstup, Casper; Cave, Chris (2021-01-15), Combining symbolic regression with the Cox proportional hazards model improves prediction of heart failure deaths, Cold Spring Harbor Laboratory, doi:10.1101/2021.01.15.21249874, S2CID 231609904
9. Christensen, Niels Johan; Demharter, Samuel; Machado, Meera; Pedersen, Lykke; Salvatore, Marco; Stentoft-Hansen, Valdemar; Iglesias, Miquel Triana (2022-06-22). "Identifying interactions in omics data for clinical biomarker discovery using symbolic regression". Bioinformatics. Oxford University Press (OUP). 38 (15): 3749–3758. doi:10.1093/bioinformatics/btac405. ISSN 1367-4803. PMC 9344843. PMID 35731214.
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Wikipedia
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QMA
In computational complexity theory, QMA, which stands for Quantum Merlin Arthur, is the set of languages for which, when a string is in the language, there is a polynomial-size quantum proof (a quantum state) that convinces a polynomial time quantum verifier (running on a quantum computer) of this fact with high probability. Moreover, when the string is not in the language, every polynomial-size quantum state is rejected by the verifier with high probability.
The relationship between QMA and BQP is analogous to the relationship between complexity classes NP and P. It is also analogous to the relationship between the probabilistic complexity class MA and BPP.
QAM is a related complexity class, in which fictional agents Arthur and Merlin carry out the sequence: Arthur generates a random string, Merlin answers with a quantum certificate and Arthur verifies it as a BQP machine.
Definition
A language L is in ${\mathsf {QMA}}(c,s)$ if there exists a polynomial time quantum verifier V and a polynomial $p(x)$ such that:[1][2][3]
• $\forall x\in L$, there exists a quantum state $|\psi \rangle $ such that the probability that V accepts the input $(|x\rangle ,|\psi \rangle )$ is greater than c.
• $\forall x\notin L$, for all quantum states $|\psi \rangle $, the probability that V accepts the input $(|x\rangle ,|\psi \rangle )$ is less than s.
where $|\psi \rangle $ ranges over all quantum states with at most $p(|x|)$ qubits.
The complexity class ${\mathsf {QMA}}$ is defined to be equal to ${\mathsf {QMA}}({2}/{3},1/3)$. However, the constants are not too important since the class remains unchanged if c and s are set to any constants such that c is greater than s. Moreover, for any polynomials $q(n)$ and $r(n)$, we have
${\mathsf {QMA}}\left({\frac {2}{3}},{\frac {1}{3}}\right)={\mathsf {QMA}}\left({\frac {1}{2}}+{\frac {1}{q(n)}},{\frac {1}{2}}-{\frac {1}{q(n)}}\right)={\mathsf {QMA}}(1-2^{-r(n)},2^{-r(n)})$.
Problems in QMA
Since many interesting classes are contained in QMA, such as P, BQP and NP, all problems in those classes are also in QMA. However, there are problems that are in QMA but not known to be in NP or BQP. Some such well known problems are discussed below.
A problem is said to be QMA-hard, analogous to NP-hard, if every problem in QMA can be reduced to it. A problem is said to be QMA-complete if it is QMA-hard and in QMA.
The local Hamiltonian problem
A k-local Hamiltonian (quantum mechanics) $H$ is a Hermitian matrix acting on n qubits which can be represented as the sum of $m$ Hamiltonian Terms acting upon at most $k$ qubits each.
$H=\sum _{i=1}^{m}H_{i}$
The general k-local Hamiltonian problem is, given a k-local Hamiltonian $H$, to find the smallest eigenvalue $\lambda $ of $H$.[4] $\lambda $ is also called the ground state energy of the Hamiltonian.
The decision version of the k-local Hamiltonian problem is a type of promise problem and is defined as, given a k-local Hamiltonian and $\alpha ,\beta $ where $\alpha >\beta $, to decide if there exists a quantum eigenstate $|\psi \rangle $ of $H$ with associated eigenvalue $\lambda $, such that $\lambda \leq \beta $ or if $\lambda \geq \alpha $.
The local Hamiltonian problem is the quantum analogue of MAX-SAT. The k-local Hamiltonian problem is QMA-complete for k ≥ 2.[5]
The 2-local Hamiltonian problem restricted to act on a two dimensional grid of qubits, is also QMA-complete.[6] It has been shown that the k-local Hamiltonian problem is still QMA-hard even for Hamiltonians representing a 1-dimensional line of particles with nearest-neighbor interactions with 12 states per particle.[7] If the system is translationally-invariant, its local Hamiltonian problem becomes QMAEXP-complete (as the problem input is encoded in the system size, the verifier now has exponential runtime while maintaining the same promise gap).[8][9]
QMA-hardness results are known for simple lattice models of qubits such as the ZX Hamiltonian [10] $H_{ZX}=\sum _{i}h_{i}Z_{i}+\sum _{i}\Delta _{i}X_{i}+\sum _{i<j}J^{ij}Z_{i}Z_{j}+\sum _{i<j}K^{ij}X_{i}X_{j}$ where $Z,X$ represent the Pauli matrices $\sigma _{z},\sigma _{x}$. Such models are applicable to universal adiabatic quantum computation.
k-local Hamiltonians problems are analogous to classical Constraint Satisfaction Problems.[11] The following table illustrates the analogous gadgets between classical CSPs and Hamiltonians.
Classical Quantum Notes
Constraint Satisfaction Problem Hamiltonian
Variable Qubit
Constraint Hamiltonian Term
Variable Assignment Quantum state
Number of constraints satisfied Hamiltonian's energy term
Optimal Solution Hamiltonian's ground state The most possible constraints satisfied
Other QMA-complete problems
A list of known QMA-complete problems can be found at https://arxiv.org/abs/1212.6312.
Related classes
QCMA (or MQA[2]), which stands for Quantum Classical Merlin Arthur (or Merlin Quantum Arthur), is similar to QMA, but the proof has to be a classical string. It is not known whether QMA equals QCMA, although QCMA is clearly contained in QMA.
QIP(k), which stands for Quantum Interactive Polynomial time (k messages), is a generalization of QMA where Merlin and Arthur can interact for k rounds. QMA is QIP(1). QIP(2) is known to be in PSPACE.[12]
QIP is QIP(k) where k is allowed to be polynomial in the number of qubits. It is known that QIP(3) = QIP.[13] It is also known that QIP = IP = PSPACE.[14]
Relationship to other classes
QMA is related to other known complexity classes by the following relations:
${\mathsf {P}}\subseteq {\mathsf {NP}}\subseteq {\mathsf {MA}}\subseteq {\mathsf {QCMA}}\subseteq {\mathsf {QMA}}\subseteq {\mathsf {PP}}\subseteq {\mathsf {PSPACE}}$
The first inclusion follows from the definition of NP. The next two inclusions follow from the fact that the verifier is being made more powerful in each case. QCMA is contained in QMA since the verifier can force the prover to send a classical proof by measuring proofs as soon as they are received. The fact that QMA is contained in PP was shown by Alexei Kitaev and John Watrous. PP is also easily shown to be in PSPACE.
It is unknown if any of these inclusions is unconditionally strict, as it is not even known whether P is strictly contained in PSPACE or P = PSPACE. However, the currently best known upper bounds on QMA are [15] [16]
${\mathsf {QMA}}\subseteq {\mathsf {A_{0}PP}}$ and ${\mathsf {QMA}}\subseteq {\mathsf {P^{QMA[log]}}}$,
where both ${\mathsf {A_{0}PP}}$ and ${\mathsf {P^{QMA[log]}}}$ are contained in ${\mathsf {PP}}$. It is unlikely that ${\mathsf {QMA}}$ equals ${\mathsf {P^{QMA[log]}}}$, as this would imply ${\mathsf {QMA}}={\mathsf {co}}$-${\mathsf {QMA}}$. It is unknown whether ${\mathsf {P^{QMA[log]}}}\subseteq {\mathsf {A_{0}PP}}$ or vice versa.
References
1. Aharonov, Dorit; Naveh, Tomer (2002). "Quantum NP – A Survey". arXiv:quant-ph/0210077v1.
2. Watrous, John (2009). "Quantum Computational Complexity". In Meyers, Robert A. (ed.). Encyclopedia of Complexity and Systems Science. pp. 7174–7201. arXiv:0804.3401. doi:10.1007/978-0-387-30440-3_428.
3. Gharibian, Sevag; Huang, Yichen; Landau, Zeph; Shin, Seung Woo (2015). "Quantum Hamiltonian Complexity". Foundations and Trends in Theoretical Computer Science. 10 (3): 159–282. arXiv:1401.3916. doi:10.1561/0400000066.
4. O'Donnel, Ryan. "Lecture 24: QMA: Quantum Merlin Arthur" (PDF). Retrieved 18 April 2021.
5. Kempe, Julia; Kitaev, Alexei; Regev, Oded (2006). "The complexity of the local Hamiltonian problem". SIAM Journal on Computing. 35 (5): 1070–1097. arXiv:quant-ph/0406180v2. doi:10.1137/S0097539704445226..
6. Oliveira, Roberto; Terhal, Barbara M. (2008). "The complexity of quantum spin systems on a two-dimensional square lattice". Quantum Information and Computation. 8 (10): 900–924. arXiv:quant-ph/0504050. Bibcode:2005quant.ph..4050O.
7. Aharonov, Dorit; Gottesman, Daniel; Irani, Sandy; Kempe, Julia (2009). "The power of quantum systems on a line". Communications in Mathematical Physics. 287 (1): 41–65. arXiv:0705.4077. Bibcode:2009CMaPh.287...41A. doi:10.1007/s00220-008-0710-3.
8. Aharonov, Dorit; Gottesman, Daniel; Irani, Sandy; Kempe, Julia (1 April 2009). "The Power of Quantum Systems on a Line". Communications in Mathematical Physics. 287 (1): 41–65. doi:10.1007/s00220-008-0710-3.
9. Bausch, Johannes; Cubitt, Toby; Ozols, Maris (November 2017). "The Complexity of Translationally Invariant Spin Chains with Low Local Dimension". Annales Henri Poincaré. 18 (11): 3449–3513. doi:10.1007/s00023-017-0609-7.
10. Biamonte, Jacob; Love, Peter (2008). "Realizable Hamiltonians for universal adiabatic quantum computers". Physical Review A. 78 (1): 012352. arXiv:0704.1287. Bibcode:2008PhRvA..78a2352B. doi:10.1103/PhysRevA.78.012352..
11. Yuen, Henry. "The Complexity of Entanglement" (PDF). henryyuen.net. Retrieved 20 April 2021.
12. Jain, Rahul; Upadhyay, Sarvagya; Watrous, John (2009). "Two-message quantum interactive proofs are in PSPACE". Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS '09). IEEE Computer Society. pp. 534–543. doi:10.1109/FOCS.2009.30. ISBN 978-0-7695-3850-1.
13. Watrous, John (2003). "PSPACE has constant-round quantum interactive proof systems". Theoretical Computer Science. 292 (3): 575–588. doi:10.1016/S0304-3975(01)00375-9.
14. Jain, Rahul; Ji, Zhengfeng; Upadhyay, Sarvagya; Watrous, John (2011). "QIP = PSPACE". Journal of the ACM. 58 (6): A30. doi:10.1145/2049697.2049704.
15. Vyalyi, Mikhail N. (2003). "QMA = PP implies that PP contains PH". Electronic Colloquium on Computational Complexity.
16. Gharibian, Sevag; Yirka, Justin (2019). "The complexity of simulating local measurements on quantum systems". Quantum. 3: 189. doi:10.22331/q-2019-09-30-189.
External links
• Aaronson, Scott. "PHYS771 Lecture 13: How Big are Quantum States?".
• Gharibian, Sevag. "Lecture 5: Quantum Merlin Arthur (QMA) and strong error reduction" (PDF).
• Complexity Zoo: QMA
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
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• No-teleportation
• PBR
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• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
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• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
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• BB84
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• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
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• Hidden subgroup
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• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
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Wikipedia
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Schur decomposition
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily equivalent to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix.
Statement
The Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as[1][2][3]
$A=QUQ^{-1}$
where Q is a unitary matrix (so that its inverse Q−1 is also the conjugate transpose Q* of Q), and U is an upper triangular matrix, which is called a Schur form of A. Since U is similar to A, it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of U.
The Schur decomposition implies that there exists a nested sequence of A-invariant subspaces {0} = V0 ⊂ V1 ⊂ ⋯ ⊂ Vn = Cn, and that there exists an ordered orthonormal basis (for the standard Hermitian form of Cn) such that the first i basis vectors span Vi for each i occurring in the nested sequence. Phrased somewhat differently, the first part says that a linear operator J on a complex finite-dimensional vector space stabilizes a complete flag (V1, ..., Vn).
Proof
A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue λ, corresponding to some eigenspace Vλ. Let Vλ⊥ be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases Z1 and Z2 spanning Vλ and Vλ⊥ respectively)
${\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}^{*}A{\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \,I_{\lambda }&A_{12}\\0&A_{22}\end{bmatrix}}:{\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}\rightarrow {\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}$
where Iλ is the identity operator on Vλ. The above matrix would be upper-triangular except for the A22 block. But exactly the same procedure can be applied to the sub-matrix A22, viewed as an operator on Vλ⊥, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since each conjugation increases the dimension of the upper-triangular block by at least one, this process takes at most n steps. Thus the space Cn will be exhausted and the procedure has yielded the desired result.[4]
The above argument can be slightly restated as follows: let λ be an eigenvalue of A, corresponding to some eigenspace Vλ. A induces an operator T on the quotient space Cn/Vλ. This operator is precisely the A22 submatrix from above. As before, T would have an eigenspace, say Wμ ⊂ Cn modulo Vλ. Notice the preimage of Wμ under the quotient map is an invariant subspace of A that contains Vλ. Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that A stabilizes.
Notes
Although every square matrix has a Schur decomposition, in general this decomposition is not unique. For example, the eigenspace Vλ can have dimension > 1, in which case any orthonormal basis for Vλ would lead to the desired result.
Write the triangular matrix U as U = D + N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A, while the Frobenius norm of A, squared, is the sum of the squared singular values of A). The nilpotent part N is generally not unique either, but its Frobenius norm is uniquely determined by A (just because the Frobenius norm of A is equal to the Frobenius norm of U = D + N).[5]
It is clear that if A is a normal matrix, then U from its Schur decomposition must be a diagonal matrix and the column vectors of Q are the eigenvectors of A. Therefore, the Schur decomposition extends the spectral decomposition. In particular, if A is positive definite, the Schur decomposition of A, its spectral decomposition, and its singular value decomposition coincide.
A commuting family {Ai} of matrices can be simultaneously triangularized, i.e. there exists a unitary matrix Q such that, for every Ai in the given family, Q Ai Q* is upper triangular. This can be readily deduced from the above proof. Take element A from {Ai} and again consider an eigenspace VA. Then VA is invariant under all matrices in {Ai}. Therefore, all matrices in {Ai} must share one common eigenvector in VA. Induction then proves the claim. As a corollary, we have that every commuting family of normal matrices can be simultaneously diagonalized.
In the infinite dimensional setting, not every bounded operator on a Banach space has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to compact operators. Every compact operator on a complex Banach space has a nest of closed invariant subspaces.
Computation
The Schur decomposition of a given matrix is numerically computed by the QR algorithm or its variants. In other words, the roots of the characteristic polynomial corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition. Conversely, the QR algorithm can be used to compute the roots of any given characteristic polynomial by finding the Schur decomposition of its companion matrix. Similarly, the QR algorithm is used to compute the eigenvalues of any given matrix, which are the diagonal entries of the upper triangular matrix of the Schur decomposition. Although the QR algorithm is formally an infinite sequence of operations, convergence to machine precision is practically achieved in ${\mathcal {O}}(n^{3})$ operations.[6] See the Nonsymmetric Eigenproblems section in LAPACK Users' Guide.[7]
Applications
Lie theory applications include:
• Every invertible operator is contained in a Borel group.
• Every operator fixes a point of the flag manifold.
Generalized Schur decomposition
Given square matrices A and B, the generalized Schur decomposition factorizes both matrices as $A=QSZ^{*}$ and $B=QTZ^{*}$, where Q and Z are unitary, and S and T are upper triangular. The generalized Schur decomposition is also sometimes called the QZ decomposition.[2]: 375
The generalized eigenvalues $\lambda $ that solve the generalized eigenvalue problem $A\mathbf {x} =\lambda B\mathbf {x} $ (where x is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of S to those of T. That is, using subscripts to denote matrix elements, the ith generalized eigenvalue $\lambda _{i}$ satisfies $\lambda _{i}=S_{ii}/T_{ii}$.
References
1. Horn, R.A. & Johnson, C.R. (1985). Matrix Analysis. Cambridge University Press. ISBN 0-521-38632-2. (Section 2.3 and further at p. 79)
2. Golub, G.H. & Van Loan, C.F. (1996). Matrix Computations (3rd ed.). Johns Hopkins University Press. ISBN 0-8018-5414-8.(Section 7.7 at p. 313)
3. Schott, James R. (2016). Matrix Analysis for Statistics (3rd ed.). New York: John Wiley & Sons. pp. 175–178. ISBN 978-1-119-09247-6.
4. Wagner, David. "Proof of Schur's Theorem" (PDF). Notes on Linear Algebra.
5. Higham, Nick. "What Is a Schur Decomposition?".
6. Trefethen, Lloyd N.; Bau, David (1997). Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics. pp. 193–194. ISBN 0-89871-361-7. OCLC 36084666.{{cite book}}: CS1 maint: date and year (link)
7. Anderson, E; Bai, Z; Bischof, C; Blackford, S; Demmel, J; Dongarra, J; Du Croz, J; Greenbaum, A; Hammarling, S; McKenny, A; Sorensen, D (1995). LAPACK Users guide. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 0-89871-447-8.
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Wikipedia
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h topology
In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes.[1][2] It combines several good properties possessed by its related "sub"topologies, such as the qfh and cdh topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmanian, Huber and Jörder's study of differential forms, etc.
Definition
Voevodsky defined the h topology to be the topology associated to finite families $\{p_{i}:U_{i}\to X\}$ of morphisms of finite type such that $\amalg U_{i}\to X$ is a universal topological epimorphism (i.e., a set of points in the target is an open subset if and only if its preimage is open, and any base change also has this property[3][4]). Voevodsky worked with this topology exclusively on categories $Sch_{/S}^{ft}$ of schemes of finite type over a Noetherian base scheme S.
Bhatt-Scholze define the h topology on the category $Sch_{/S}^{fp}$ of schemes of finite presentation over a qcqs base scheme $S$ to be generated by $v$-covers of finite presentation. They show (generalising results of Voevodsky) that the h topology is generated by:
1. fppf-coverings, and
2. families of the form $\{X'\to X,Z\to X\}$ where
1. $X'\to X$ is a proper morphism of finite presentation,
2. $Z\to X$ is a closed immersion of finite presentation, and
3. $X'\to X$ is an isomorphism over $X\setminus Z$.
Note that $X'=\varnothing $ is allowed in an abstract blowup, in which case Z is a nilimmersion of finite presentation.
Examples
The h-topology is not subcanonical, so representable presheaves are almost never h-sheaves. However, the h-sheafification of representable sheaves are interesting and useful objects; while presheaves of relative cycles are not representable, their associated h-sheaves are representable in the sense that there exists a disjoint union of quasi-projective schemes whose h-sheafifications agree with these h-sheaves of relative cycles.[5]
Any h-sheaf in positive characteristic satisfies $F(X)=F(X^{perf})$ where we interpret $X^{perf}$ as the colimit $\operatorname {colim} (F(X){\stackrel {\text{Frob}}{\to }}F(X){\stackrel {\text{Frob}}{\to }}\dots )$ over the Frobenii (if the Frobenius is of finite presentation, and if not, use an analogous colimit consisting of morphisms of finite presentation). In fact, (in positive characteristic) the h-sheafification ${\mathcal {O}}_{h}$ of the structure sheaf ${\mathcal {O}}$ is given by ${\mathcal {O}}_{h}(X)={\mathcal {O}}(X^{perf})$. So the structure sheaf "is an h-sheaf on the category of perfect schemes" (although this sentence doesn't really make sense mathematically since morphisms between perfect schemes are almost never of finite presentation). In characteristic zero similar results hold with perfection replaced by semi-normalisation.
Huber-Jörder study the h-sheafification $\Omega _{h}^{n}$ of the presheaf $X\mapsto \Gamma (X,\Omega _{X/k}^{n})$ of Kähler differentials on categories of schemes of finite type over a characteristic zero base field $k$. They show that if X is smooth, then $\Omega _{h}^{n}(X)=\Gamma (X,\Omega _{X/k}^{n})$, and for various nice non-smooth X, the sheaf $\Omega _{h}^{n}$ recovers objects such as reflexive differentials and torsion-free differentials. Since the Frobenius is an h-covering, in positive characteristic we get $\Omega _{h}^{n}=0$ for $n>0$, but analogous results are true if we replace the h-topology with the cdh-topology.
By the Nullstellensatz, a morphism of finite presentation $X\to \operatorname {Spec} (k)$ towards the spectrum of a field $k$ admits a section up to finite extension. That is, there exists a finite field extension $L/k$ and a factorisation $\operatorname {Spec} (L)\to X\to \operatorname {Spec} (k)$. Consequently, for any presheaf $F$ and field $k$ we have $F_{h}(k)=F_{et}(k^{perf})$ where $F_{h}$, resp. $F_{et}$, denotes the h-sheafification, resp. etale sheafification.
Properties
As mentioned above, in positive characteristic, any h-sheaf satisfies $F(X)=F(X^{perf})$. In characteristic zero, we have $F(X)=F(X^{sn})$ where $X^{sn}$ is the semi-normalisation (the scheme with the same underlying topological space, but the structure sheaf is replaced with its termwise seminormalisation).
Since the h-topology is finer than the Zariski topology, every scheme admits an h-covering by affine schemes.
Using abstract blowups and Noetherian induction, if $k$ is a field admitting resolution of singularities (e.g., a characteristic zero field) then any scheme of finite type over $k$ admits an h-covering by smooth $k$-schemes. More generally, in any situation where de Jong's theorem on alterations is valid we can find h-coverings by regular schemes.
Since finite morphisms are h-coverings, algebraic correspondences are finite sums of morphisms.[6]
cdh topology
The cdh topology on the category $Sch_{/S}^{fp}$ of schemes of finite presentation over a qcqs base scheme $S$ is generated by:
1. Nisnevich coverings, and
2. families of the form $\{X'\to X,Z\to X\}$ where
1. $X'\to X$ is a proper morphism of finite presentation,
2. $Z\to X$ is a closed immersion of finite presentation, and
3. $X'\to X$ is an isomorphism over $X\setminus Z$.
The cd stands for completely decomposed (in the same sense it is used for the Nisnevich topology). As mentioned in the examples section, over a field admitting resolution of singularities, any variety admits a cdh-covering by smooth varieties. This topology is heavily used in the study of Voevodsky motives with integral coefficients (with rational coefficients the h-topology together with de Jong alterations is used).
Since the Frobenius is not a cdh-covering, the cdh-topology is also a useful replacement for the h-topology in the study of differentials in positive characteristic.
Rather confusingly, there are completely decomposed h-coverings, which are not cdh-coverings, for example the completely decomposed family of flat morphisms $\{\mathbb {A} ^{1}{\stackrel {x\mapsto x^{2}}{\to }}\mathbb {A} ^{1},\mathbb {A} ^{1}\setminus \{0\}{\stackrel {x\mapsto x}{\to }}\mathbb {A} ^{1}\}$.
Relation to v-topology and arc-topology
The v-topology (or universally subtrusive topology) is equivalent to the h-topology on the category $Sch_{S}^{ft}$ of schemes of finite type over a Noetherian base scheme S. Indeed, a morphism in $Sch_{S}^{ft}$ is universally subtrusive if and only if it is universally submersive Rydh (2010, Cor.2.10). In other words,
$Shv_{h}(Sch_{S}^{ft})=Shv_{v}(Sch_{S}^{ft}),\qquad (S\ {\textrm {Noetherian}})$
More generally, on the category $Sch$ of all qcqs schemes, neither of the v- nor the h- topologies are finer than the other: $Shv_{h}(Sch)\not \subset Shv_{v}(Sch)$ and $Shv_{v}(Sch)\not \subset Shv_{h}(Sch)$. There are v-covers which are not h-covers (e.g., $Spec(\mathbb {C} (x))\to Spec(\mathbb {C} )$) and h-covers which are not v-covers (e.g., $Spec(R/{\mathfrak {p}})\sqcup Spec(R_{\mathfrak {p}})\to Spec(R)$ where R is a valuation ring of rank 2 and ${\mathfrak {p}}$ is the non-open, non-closed prime Rydh (2010, Example 4.3)).
However, we could define an h-analogue of the fpqc topology by saying that an hqc-covering is a family $\{T_{i}\to T\}_{i\in I}$ such that for each affine open $U\subseteq T$ there exists a finite set K, a map $i:K\to I$ and affine opens $U_{i(k)}\subseteq T_{i(k)}\times _{T}U$ such that $\sqcup _{k\in K}U_{i(k)}\to U$ is universally submersive (with no finiteness conditions). Then every v-covering is an hqc-covering.
$Shv_{hqc}(Sch)\subsetneq Shv_{v}(Sch).$
Indeed, any subtrusive morphism is submersive (this is an easy exercise using Rydh (2010, Cor.1.5 and Def.2.2)).
By a theorem of Rydh, for a map $f:Y\to X$ of qcqs schemes with $X$ Noetherian, $f$ is a v-cover if and only if it is an arc-cover (for the statement in this form see Bhatt & Mathew (2018, Prop.2.6)). That is, in the Noetherian setting everything said above for the v-topology is valid for the arc-topology.
Notes
1. Voevodsky. "Homology of schemes, I". {{cite journal}}: Cite journal requires |journal= (help)
2. Suslin, Voevodsky. "Singular homology of abstract algebraic varieties". {{cite journal}}: Cite journal requires |journal= (help)
3. SGA I, Exposé IX, définition 2.1
4. Suslin and Voevodsky, 4.1
5. 1=Suslin 2=Voevodsky. "Relative cycles". {{cite journal}}: Cite journal requires |journal= (help)
6. Suslin, Voevodsky, Singular homology of abstract algebraic varieties
References
• Suslin, A., and Voevodsky, V., Relative cycles and Chow sheaves, April 1994, .
• Bhatt, Bhargav; Mathew, Akhil (2018), The arc-topology, arXiv:1807.04725v2
• Rydh, David (2010), "Submersions and effective descent of étale morphisms", Bull. Soc. Math. France, 138 (2): 181–230, arXiv:0710.2488, doi:10.24033/bsmf.2588, MR 2679038, S2CID 17484591
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Wikipedia
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Qi-Man Shao
Qi-Man Shao (Chinese: 邵启满; born 1962) is a Chinese probabilist and statistician mostly known for his contributions to asymptotic theory in probability and statistics. He is currently a Chair Professor of Statistics and Data Science[1] at the Southern University of Science and Technology.
Biography
He earned a bachelor's degree in Mathematics and a master's degree in Statistics & Probability from Hangzhou University (now Zhejiang University) in 1983 and 1986, respectively. He went to graduate school at the University of Science and Technology of China and received a Ph.D. degree in Statistics & Probability in 1989. He spent four years as lecturer and then associate professor at Hangzhou University from 1986 to 1990. In July 1990, he joined Carleton University, Canada as a visiting research fellow, working with Csörgő Miklós. From September 1991 to August 1992, he worked as a Taft Postdoctoral Fellow at the University of Cincinnati. He joined the National University of Singapore as a lecturer in 1992, and later become a senior lecturer.
He joined the University of Oregon as an assistant professor in 1996, and was later promoted to associate professor and professor. From 2005 to 2012, he was a professor and Chair Professor at the Hong Kong University of Science and Technology. In 2012, he moved to the Chinese University of Hong Kong, where he served as Department Chair from 2013 to 2018[2] and became the Choh-Ming Li Professor of Statistics in 2015.[3] Starting March 2019, he moved to the Southern University of Science and Technology, as a Chair Professor and the Founding Chairman of the Department of Statistics and Data Science.
His research interests include asymptotic theory in probability and statistics, self-normalized limit theory, Stein’s method, high-dimensional and large-scale statistical analysis. He is particularly well-known for his fundamental contributions to self-normalized large and moderate deviation theories, Stein’s method for normal and non-normal approximation, and the development of various probability inequalities for dependent random variables. He authored and co-authored over 180 articles on probability and statistics, and co-authored three well-known books (Monte Carlo Methods in Bayesian Computation (2000),[4] Self-normalized Processes: Limit Theory and Statistical Applications (2009),[5] and Normal Approximation by Stein’s Method (2011)[6]).
Honors and awards
• Fok Ying Tung Education Foundation Award, 1989
• The State Natural Science Award (the 3rd class), 1997 (Z.Y. Lin, C.R. Lu and Q.M. Shao)
• Elected Fellow, the Institute of Mathematical Statistics,[7] 2001
• Invited speaker (45min) at the 2010 International Congress of Mathematicians[8]
• IMS Medallion Lecturer, Keynote Speaker at the 2011 Joint Statistical Meetings[9]
• Plenary speaker, 36th Conference on Stochastic Processes and Their Applications, 2013[10]
• Plenary speaker, IMS-China International Conference on Statistics and Probability, 2013
• The State Natural Science Award (the 2nd class), 2015 (Q.-M. Shao and B.-Y. Jing)[11]
Professional services
• co-Editor, The Annals of Applied Probability (1/2022 - 12/2024)[12]
• Associate Editor, Bernoulli (1/2013 - 12/2021)
• Associate Editor-in-Chief, Science China: Mathematics (1/2013 - 12/2027)
• Associate Editor, The Annals of Statistics (11/2003 - 12/2012)
• Associate Editor, The Annals of Applied Probability (1/2006 -12/2012)
• Associate Editor, ESAIM-P&S (1/2006-12/2021)
• Institute of Mathematical Statistics (IMS) Committee on Fellows, Member in 2007-2009 and 2011, Chair in 2009
• IMS Committee on Nominations (2011, 2016, 2017), Institute of Mathematical Statistics[13]
• Chair, Local Organizing Committee, the 4th IMS Asian Pacific Rim Meeting, 2016[14]
• co-Chair, Scientific Program Committee, the 5th IMS Asian Pacific Rim Meeting, 2018[13]
• The Scientific Program Committee, The World Congress in Probability and Statistics, 2008,[15] 2020<[16]
• Council Member, Institute of Mathematical Statistics (2019-2022)[17]
References
1. "Faculty Profiles - SUSTech". faculty.sustech.edu.cn. Retrieved 2022-12-16.
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Shing-Tung Yau
Shing-Tung Yau (/jaʊ/; Chinese: 丘成桐; pinyin: Qiū Chéngtóng; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.[1] In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University.[2]
Not to be confused with his brother, Stephen Shing-Toung Yau.
Shing-Tung Yau
Born (1949-04-04) April 4, 1949
Shantou, Guangdong, China
NationalityUnited States (since 1990)
Alma materChinese University of Hong Kong (1966–69)
University of California, Berkeley (Ph.D. 1971)
Known for
• Analysis and geometry of the Plateau problem and minimal surfaces
• Bernstein's problem for maximal surfaces
• Bogomolov–Miyaoka–Yau inequality
• Donaldson–Uhlenbeck–Yau theorem
• Frankel conjecture
• Geometry of positive scalar curvature
• Gradient estimates for partial differential equations
• Minkowski problem
• Monge–Ampère equation
• Omori−Yau maximum principle
• Positive mass theorem
• Resolution of the Calabi conjecture and construction of Calabi–Yau manifolds
• Resolution of the Willmore conjecture in the non-embedded case
• SYZ conjecture and mirror symmetry
• Thomas–Yau conjecture
SpouseYu-yun Kuo
ChildrenMichael Yau, Isaac Yau
AwardsJohn J. Carty Award (1981)
Veblen Prize (1981)
Fields Medal (1982)
Crafoord Prize (1994)
National Medal of Science (1997)
Wolf Prize (2010)
Shaw Prize (2023)
Scientific career
FieldsMathematics
InstitutionsTsinghua University
Harvard University
Stanford University
Stony Brook University
Institute for Advanced Study
University of California, San Diego
ThesisOn the Fundamental Group of Compact Manifolds of Non-Positive Curvature (1971)
Doctoral advisorShiing-Shen Chern
Doctoral studentsRichard Schoen (Stanford, 1977)
Robert Bartnik (Princeton, 1983)
Mark Stern (Princeton, 1984)
Huai-Dong Cao (Princeton, 1986)
Gang Tian (Harvard, 1988)
Jun Li (Stanford, 1989)
Wanxiong Shi (Harvard, 1990)
Lizhen Ji (Northeastern, 1991)
Kefeng Liu (Harvard, 1993)
Mu-Tao Wang (Harvard, 1998)
Chiu-Chu Melissa Liu (Harvard, 2002)
Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation.[3] Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied mathematics, engineering, and numerical analysis.
Biography
Yau was born in Shantou, China in 1949 to Hakka parents. Yau's ancestral hometown is Jiaoling county, China. His mother, Yeuk Lam Leung, was from Meixian District; his father, Chen Ying Chiu, was a Chinese scholar of philosophy, history, literature, and economics.[YN19] He was the fifth of eight children, with Hakka ancestry.[4]
During the Communist takeover of mainland China, when he was only a few months old, his family moved to Hong Kong where he was forced to learn to speak the Cantonese language as well as speak the Chinese dialect of Hakka. He was not able to revisit until 1979, at the invitation of Hua Luogeng, when mainland China entered the reform and opening era.[YN19]. They had financial troubles from having lost all of their possessions, and his father and second-oldest sister died when he was thirteen. Yau began to read and appreciate his father's books, and became more devoted to schoolwork. After graduating from Pui Ching Middle School, he studied mathematics at the Chinese University of Hong Kong from 1966 to 1969, without receiving a degree due to graduating early. He left his textbooks with his younger brother, Stephen Shing-Toung Yau, who then decided to major in mathematics as well.
Yau left for the Ph.D. program in mathematics at University of California, Berkeley in the fall of 1969. Over the winter break, he read the first issues of the Journal of Differential Geometry, and was deeply inspired by John Milnor's papers on geometric group theory.[5][YN19] Subsequently he formulated a generalization of Preissman's theorem, and developed his ideas further with Blaine Lawson over the next semester.[6] Using this work, he received his Ph.D. the following year, in 1971, under the supervision of Shiing-Shen Chern.[7]
He spent a year as a member of the Institute for Advanced Study at Princeton before joining Stony Brook University in 1972 as an assistant professor. In 1974, he became an associate professor at Stanford University.[8] In 1976 he took a visiting faculty position with UCLA and married physicist Yu-Yun Kuo, who he knew from his time as a graduate student at Berkeley. From 1984 to 1987 he worked at University of California, San Diego.[9] Since 1987, he has been at Harvard University.[10] In April 2022, Yau announced a forthcoming move from Harvard to Tsinghua University.[2]
In 1978, Yau became "stateless" after the British Consulate revoked his Hong Kong residency due to his United States permanent residency status.[11][12] Regarding his status when receiving his Fields Medal in 1982, Yau stated "I am proud to say that when I was awarded the Fields Medal in mathematics, I held no passport of any country and should certainly be considered Chinese."[13] Yau remained "stateless" until 1990, when he obtained United States citizenship.[11][14]
With science journalist Steve Nadis, Yau has written a non-technical account of Calabi-Yau manifolds and string theory,[YN10] a history of Harvard's mathematics department,[NY13] and an autobiography.[YN19]
Academic activities
Yau has made major contributions to the development of modern differential geometry and geometric analysis. As said by William Thurston in 1981:[15]
We have rarely had the opportunity to witness the spectacle of the work of one mathematician affecting, in a short span of years, the direction of whole areas of research. In the field of geometry, one of the most remarkable instances of such an occurrence during the last decade is given by the contributions of Shing-Tung Yau.
His most widely celebrated results include the resolution (with Shiu-Yuen Cheng) of the boundary-value problem for the Monge-Ampère equation, the positive mass theorem in the mathematical analysis of general relativity (achieved with Richard Schoen), the resolution of the Calabi conjecture, the topological theory of minimal surfaces (with William Meeks), the Donaldson-Uhlenbeck-Yau theorem (done with Karen Uhlenbeck), and the Cheng−Yau and Li−Yau gradient estimates for partial differential equations (found with Shiu-Yuen Cheng and Peter Li). Many of Yau's results (in addition to those of others) were written into textbooks co-authored with Schoen.[SY94][SY97]
In addition to his research, Yau is the founder and director of several mathematical institutes, mostly in China. John Coates has commented that "no other mathematician of our times has come close" to Yau's success at fundraising for mathematical activities in China and Hong Kong.[6] During a sabbatical year at National Tsinghua University in Taiwan, Yau was asked by Charles Kao to start a mathematical institute at the Chinese University of Hong Kong. After a few years of fundraising efforts, Yau established the multi-disciplinary Institute of Mathematical Sciences in 1993, with his frequent co-author Shiu-Yuen Cheng as associate director. In 1995, Yau assisted Yongxiang Lu with raising money from Ronnie Chan and Gerald Chan's Morningside Group for the new Morningside Center of Mathematics at the Chinese Academy of Sciences. Yau has also been involved with the Center of Mathematical Sciences at Zhejiang University,[16] at Tsinghua University,[17] at National Taiwan University,[18] and in Sanya.[19] More recently, in 2014, Yau raised money to establish the Center of Mathematical Sciences and Applications (of which he is the director), the Center for Green Buildings and Cities, and the Center for Immunological Research, all at Harvard University.[20]
Modeled on an earlier physics conference organized by Tsung-Dao Lee and Chen-Ning Yang, Yau proposed the International Congress of Chinese Mathematicians, which is now held every three years. The first congress was held at the Morningside Center from December 12 to 18, 1998. He co-organizes the annual "Journal of Differential Geometry" and "Current Developments in Mathematics" conferences. Yau is an editor-in-chief of the Journal of Differential Geometry,[21] Asian Journal of Mathematics,[22] and Advances in Theoretical and Mathematical Physics.[23] As of 2021, he has advised over seventy Ph.D. students.[7]
In Hong Kong, with the support of Ronnie Chan, Yau set up the Hang Lung Award for high school students. He has also organized and participated in meetings for high school and college students, such as the panel discussions Why Math? Ask Masters! in Hangzhou, July 2004, and The Wonder of Mathematics in Hong Kong, December 2004. Yau also co-initiated a series of books on popular mathematics, "Mathematics and Mathematical People".
In 2002 and 2003, Grigori Perelman posted preprints to the arXiv claiming to prove the Thurston geometrization conjecture and, as a special case, the renowned Poincaré conjecture. Although his work contained many new ideas and results, his proofs lacked detail on a number of technical arguments. Over the next few years, several mathematicians devoted their time to fill in details and provide expositions of Perelman's work to the mathematical community.[24] A well-known August 2006 article in the New Yorker written by Sylvia Nasar and David Gruber about the situation brought some professional disputes involving Yau to public attention.[13][14]
• Alexander Givental alleged that Bong Lian, Kefeng Liu, and Yau illegitimately took credit from him for resolving a well-known conjecture in the field of mirror symmetry. Although it is undisputed that Lian−Liu−Yau's article appeared after Givental's, they claim that his work contained gaps that were only filled in following work in their own publication; Givental claims that his original work was complete. Nasar and Gruber quote an anonymous mathematician as agreeing with Givental.[25]
• In the 1980s, Yau's colleague Yum-Tong Siu accused Yau's Ph.D. student Gang Tian of plagiarizing some of his work. At the time, Yau defended Tian against Siu's accusations.[YN19] In the 2000s, Yau began to amplify Siu's allegations, saying that he found Tian's dual position at Princeton University and Peking University to be highly unethical due to his high salary from Peking University compared to other professors and students who made more active contributions to the university.[26][YN19] Science Magazine covered the broader phenomena of such positions in China, with Tian and Yau as central figures.[27]
• Nasar and Gruber say that, having allegedly not done any notable work since the middle of the 1980s, Yau tried to regain prominence by claiming that Xi-Ping Zhu and Yau's former student Huai-Dong Cao had solved the Thurston and Poincaré conjectures, only partially based on some of Perelman's ideas. Nasar and Gruber quoted Yau as agreeing with the acting director of one of Yau's mathematical centers, who at a press conference assigned Cao and Zhu thirty percent of the credit for resolving the conjectures, with Perelman receiving only twenty-five (with the rest going to Richard Hamilton). A few months later, a segment of NPR's All Things Considered covering the situation reviewed an audio recording of the press conference and found no such statement made by either Yau or the acting director.[28]
Yau claimed that Nasar and Gruber's article was defamatory and contained several falsehoods, and that they did not give him the opportunity to represent his own side of the disputes. He considered filing a lawsuit against the magazine, claiming professional damage, but says he decided that it wasn't sufficiently clear what such an action would achieve.[YN19] He established a public relations website, with letters responding to the New Yorker article from several mathematicians, including himself and two others quoted in the article.[29]
In his autobiography, Yau said that his statements in 2006 such as that Cao and Zhu gave "the first complete and detailed account of the proof of the Poincaré conjecture" should have been phrased more carefully. Although he does believe Cao and Zhu's work to be the first and most rigorously detailed account of Perelman's work, he says he should have clarified that they had "not surpassed Perelman's work in any way."[YN19] He has also maintained the view that (as of 2019) the final parts of Perelman's proof should be better understood by the mathematical community, with the corresponding possibility that there remain some unnoticed errors.
Technical contributions to mathematics
Yau has made a number of major research contributions, centered on differential geometry and its appearance in other fields of mathematics and science. In addition to his research, Yau has compiled influential sets of open problems in differential geometry, including both well-known old conjectures with new proposals and problems. Two of Yau's most widely cited problem lists from the 1980s have been updated with notes on progress as of 2014.[30] Particularly well-known are a conjecture on existence of minimal hypersurfaces and on the spectral geometry of minimal hypersurfaces.
Calabi conjecture
Further information: Calabi conjecture
In 1978, by studying the complex Monge–Ampère equation, Yau resolved the Calabi conjecture, which had been posed by Eugenio Calabi in 1954.[Y78a] As a special case, this showed that Kähler-Einstein metrics exist on any closed Kähler manifold whose first Chern class is nonpositive. Yau's method adapted earlier work of Calabi, Jürgen Moser, and Aleksei Pogorelov, developed for quasilinear elliptic partial differential equations and the real Monge–Ampère equation, to the setting of the complex Monge–Ampère equation.[31][32][33][34]
• In differential geometry, Yau's theorem is significant in proving the general existence of closed manifolds of special holonomy; any simply-connected closed Kähler manifold which is Ricci flat must have its holonomy group contained in the special unitary group, according to the Ambrose–Singer theorem.[35] Examples of compact Riemannian manifolds with other special holonomy groups have been found by Dominic Joyce and Peter Kronheimer, although no proposals for general existence results, analogous to Calabi's conjecture, have been successfully identified in the case of the other groups.[32]
• In algebraic geometry, the existence of canonical metrics as proposed by Calabi allows one to give equally canonical representatives of characteristic classes by differential forms. Due to Yau's initial efforts at disproving the Calabi conjecture by showing that it would lead to contradictions in such contexts, he was able to draw striking corollaries to the conjecture itself.[Y77] In particular, the Calabi conjecture implies the Miyaoka–Yau inequality on Chern numbers of surfaces, in addition to homotopical characterizations of the complex structures of the complex projective plane and of quotients of the two-dimensional complex unit ball.[31][35]
• A special case of the Calabi conjecture asserts that a Kähler metric of zero Ricci curvature must exist on any Kähler manifold whose first Chern class is zero.[31] In string theory, it was discovered in 1985 by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten that these Calabi–Yau manifolds, due to their special holonomy, are the appropriate configuration spaces for superstrings. For this reason, Yau's resolution of the Calabi conjecture is considered to be of fundamental importance in modern string theory.[36][37][38]
The understanding of the Calabi conjecture in the noncompact setting is less definitive. Gang Tian and Yau extended Yau's analysis of the complex Monge−Ampère equation to the noncompact setting, where the use of cutoff functions and corresponding integral estimates necessitated the conditional assumption of certain controlled geometry near infinity.[TY90] This reduces the problem to the question of existence of Kähler metrics with such asymptotic properties; they obtained such metrics for certain smooth quasi-projective complex varieties. They later extended their work to allow orbifold singularities.[TY91] With Brian Greene, Alfred Shapere, and Cumrun Vafa, Yau introduced an ansatz for a Kähler metric on the set of regular points of certain surjective holomorphic maps, with Ricci curvature approximately zero.[G+90] They were able to apply the Tian−Yau existence theorem to construct a Kähler metric which is exactly Ricci-flat. The Greene−Shapere−Vafa−Yau ansatz and its natural generalization, now known as a semi-flat metric, has become important in several analyses of problems in Kähler geometry.[39][40]
Scalar curvature and general relativity
The positive energy theorem, obtained by Yau in collaboration with his former doctoral student Richard Schoen, can be described in physical terms:
In Einstein's theory of general relativity, the gravitational energy of an isolated physical system is nonnegative.
However, it is a precise theorem of differential geometry and geometric analysis, in which physical systems are modeled by Riemannian manifolds with nonnegativity of a certain generalized scalar curvature. As such, Schoen and Yau's approach originated in their study of Riemannian manifolds of positive scalar curvature, which is of interest in and of itself. The starting point of Schoen and Yau's analysis is their identification of a simple but novel way of inserting the Gauss–Codazzi equations into the second variation formula for the area of a stable minimal hypersurface of a three-dimensional Riemannian manifold. The Gauss–Bonnet theorem then highly constrains the possible topology of such a surface when the ambient manifold has positive scalar curvature.[SY79a][41][42]
Schoen and Yau exploited this observation by finding novel constructions of stable minimal hypersurfaces with various controlled properties.[SY79a] Some of their existence results were developed simultaneously with similar results of Jonathan Sacks and Karen Uhlenbeck, using different techniques. Their fundamental result is on the existence of minimal immersions with prescribed topological behavior. As a consequence of their calculation with the Gauss–Bonnet theorem, they were able to conclude that certain topologically distinguished three-dimensional manifolds cannot have any Riemannian metric of nonnegative scalar curvature.[43][44]
Schoen and Yau then adapted their work to the setting of certain Riemannian asymptotically flat initial data sets in general relativity. They proved that negativity of the mass would allow one to invoke the Plateau problem to construct stable minimal surfaces which are geodesically complete. A noncompact analogue of their calculation with the Gauss–Bonnet theorem then provides a logical contradiction to the negativity of mass. As such, they were able to prove the positive mass theorem in the special case of their Riemannian initial data sets.[SY79c][45]
Schoen and Yau extended this to the full Lorentzian formulation of the positive mass theorem by studying a partial differential equation proposed by Pong-Soo Jang. They proved that solutions to the Jang equation exist away from the apparent horizons of black holes, at which solutions can diverge to infinity.[SY81] By relating the geometry of a Lorentzian initial data set to the geometry of the graph of such a solution to the Jang equation, interpreting the latter as a Riemannian initial data set, Schoen and Yau proved the full positive energy theorem.[45] Furthermore, by reverse-engineering their analysis of the Jang equation, they were able to establish that any sufficient concentration of energy in general relativity must be accompanied by an apparent horizon.[SY83]
Due to the use of the Gauss–Bonnet theorem, these results were originally restricted to the case of three-dimensional Riemannian manifolds and four-dimensional Lorentzian manifolds. Schoen and Yau established an induction on dimension by constructing Riemannian metrics of positive scalar curvature on minimal hypersurfaces of Riemannian manifolds which have positive scalar curvature.[SY79b] Such minimal hypersurfaces, which were constructed by means of geometric measure theory by Frederick Almgren and Herbert Federer, are generally not smooth in large dimensions, so these methods only directly apply up for Riemannian manifolds of dimension less than eight. Without any dimensional restriction, Schoen and Yau proved the positive mass theorem in the class of locally conformally flat manifolds.[SY88][31] In 2017, Schoen and Yau published a preprint claiming to resolve these difficulties, thereby proving the induction without dimensional restriction and verifying the Riemannian positive mass theorem in arbitrary dimension.
Gerhard Huisken and Yau made a further study of the asymptotic region of Riemannian manifolds with strictly positive mass. Huisken had earlier initiated the study of volume-preserving mean curvature flow of hypersurfaces of Euclidean space.[46] Huisken and Yau adapted his work to the Riemannian setting, proving a long-time existence and convergence theorem for the flow. As a corollary, they established a new geometric feature of positive-mass manifolds, which is that their asymptotic regions are foliated by surfaces of constant mean curvature.[HY96]
Omori−Yau maximum principle
Traditionally, the maximum principle technique is only applied directly on compact spaces, as maxima are then guaranteed to exist. In 1967, Hideki Omori found a novel maximum principle which applies on noncompact Riemannian manifolds whose sectional curvatures are bounded below. It is trivial that approximate maxima exist; Omori additionally proved the existence of approximate maxima where the values of the gradient and second derivatives are suitably controlled. Yau partially extended Omori's result to require only a lower bound on Ricci curvature; the result is known as the Omori−Yau maximum principle.[Y75b] Such generality is useful due to the appearance of Ricci curvature in the Bochner formula, where a lower bound is also typically used in algebraic manipulations. In addition to giving a very simple proof of the principle itself, Shiu-Yuen Cheng and Yau were able to show that the Ricci curvature assumption in the Omori−Yau maximum principle can be replaced by the assumption of the existence of cutoff functions with certain controllable geometry.[CY75][31][47][48][49]
Yau was able to directly apply the Omori−Yau principle to generalize the classical Schwarz−Pick lemma of complex analysis. Lars Ahlfors, among others, had previously generalized the lemma to the setting of Riemann surfaces. With his methods, Yau was able to consider the setting of a mapping from a complete Kähler manifold (with a lower bound on Ricci curvature) to a Hermitian manifold with holomorphic bisectional curvature bounded above by a negative number.[Y78b][35][49]
Cheng and Yau extensively used their variant of the Omori−Yau principle to find Kähler−Einstein metrics on noncompact Kähler manifolds, under an ansatz developed by Charles Fefferman. The estimates involved in the method of continuity were not as difficult as in Yau's earlier work on the Calabi conjecture, due to the fact that Cheng and Yau only considered Kähler−Einstein metrics with negative scalar curvature. The more subtle question, where Fefferman's earlier work became important, is to do with geodesic completeness. In particular, Cheng and Yau were able to find complete Kähler-Einstein metrics of negative scalar curvature on any bounded, smooth, and strictly pseudoconvex subset of complex Euclidean space.[CY80] These can be thought of as complex geometric analogues of the Poincaré ball model of hyperbolic space.[35][50]
Differential Harnack inequalities
Yau's original application of the Omori−Yau maximum principle was to establish gradient estimates for a number of second-order elliptic partial differential equations.[Y75b] Given a function on a complete and smooth Riemannian manifold which satisfies various conditions relating the Laplacian to the function and gradient values, Yau applied the maximum principle to various complicated composite expressions to control the size of the gradient. Although the algebraic manipulations involved are complex, the conceptual form of Yau's proof is strikingly simple.[51][47]
Yau's novel gradient estimates have come to be called "differential Harnack inequalities" since they can be integrated along arbitrary paths in to recover inequalities which are of the form of the classical Harnack inequalities, directly comparing the values of a solution to a differential equation at two different input points. By making use of Calabi's study of the distance function on a Riemannian manifold, Yau and Shiu-Yuen Cheng gave a powerful localization of Yau's gradient estimates, using the same methods to simplify the proof of the Omori−Yau maximum principle.[CY75] Such estimates are widely quoted in the particular case of harmonic functions on a Riemannian manifold, although Yau and Cheng−Yau's original results cover more general scenarios.[51][47]
In 1986, Yau and Peter Li made use of the same methods to study parabolic partial differential equations on Riemannian manifolds.[LY86][47] Richard Hamilton generalized their results in certain geometric settings to matrix inequalities. Analogues of the Li−Yau and Hamilton−Li−Yau inequalities are of great importance in the theory of Ricci flow, where Hamilton proved a matrix differential Harnack inequality for the curvature operator of certain Ricci flows, and Grigori Perelman proved a differential Harnack inequality for the solutions of a backwards heat equation coupled with a Ricci flow.[52][51]
Cheng and Yau were able to use their differential Harnack estimates to show that, under certain geometric conditions, closed submanifolds of complete Riemannian or pseudo-Riemannian spaces are themselves complete. For instance, they showed that if M is a spacelike hypersurface of Minkowski space which is topologically closed and has constant mean curvature, then the induced Riemannian metric on M is complete.[CY76a] Analogously, they showed that if M is an affine hypersphere of affine space which is topologically closed, then the induced affine metric on M is complete.[CY86] Such results are achieved by deriving a differential Harnack inequality for the (squared) distance function to a given point and integrating along intrinsically defined paths.
Donaldson−Uhlenbeck−Yau theorem
Further information: Hermitian Yang–Mills connection
In 1985, Simon Donaldson showed that, over a nonsingular projective variety of complex dimension two, a holomorphic vector bundle admits a hermitian Yang–Mills connection if and only if the bundle is stable. A result of Yau and Karen Uhlenbeck generalized Donaldson's result to allow a compact Kähler manifold of any dimension.[UY86] The Uhlenbeck–Yau method relied upon elliptic partial differential equations while Donaldson's used parabolic partial differential equations, roughly in parallel to Eells and Sampson's epochal work on harmonic maps. The results of Donaldson and Uhlenbeck–Yau have since been extended by other authors. Uhlenbeck and Yau's article is important in giving a clear reason that stability of the holomorphic vector bundle can be related to the analytic methods used in constructing a hermitian Yang–Mills connection. The essential mechanism is that if an approximating sequence of hermitian connections fails to converge to the required Yang–Mills connection, then they can be rescaled to converge to a subsheaf which can be verified to be destabilizing by Chern–Weil theory.[33][53]
Like the Calabi–Yau theorem, the Donaldson–Uhlenbeck–Yau theorem is of interest in theoretical physics.[37] In the interest of an appropriately general formulation of supersymmetry, Andrew Strominger included the hermitian Yang–Mills condition as part of his Strominger system, a proposal for the extension of the Calabi−Yau condition to non-Kähler manifolds.[36] Ji-Xiang Fu and Yau introduced an ansatz for the solution of Strominger's system on certain three-dimensional complex manifolds, reducing the problem to a complex Monge−Ampère equation, which they solved.[FY08]
Yau's solution of the Calabi conjecture had given a reasonably complete answer to the question of how Kähler metrics on compact complex manifolds of nonpositive first Chern class can be deformed into Kähler–Einstein metrics.[Y78a] Akito Futaki showed that the existence of holomorphic vector fields can act as an obstruction to the direct extension of these results to the case when the complex manifold has positive first Chern class.[35] A proposal of Calabi's suggested that Kähler–Einstein metrics exist on any compact Kähler manifolds with positive first Chern class which admit no holomorphic vector fields.[Y82b] During the 1980s, Yau and others came to understand that this criterion could not be sufficient. Inspired by the Donaldson−Uhlenbeck−Yau theorem, Yau proposed that the existence of Kähler–Einstein metrics must be linked to stability of the complex manifold in the sense of geometric invariant theory, with the idea of studying holomorphic vector fields along projective embeddings, rather than holomorphic vector fields on the manifold itself.[Y93][Y14a] Subsequent research of Gang Tian and Simon Donaldson refined this conjecture, which became known as the Yau–Tian–Donaldson conjecture relating Kähler–Einstein metrics and K-stability. In 2019, Xiuxiong Chen, Donaldson, and Song Sun were awarded the Oswald Veblen prize for resolution of the conjecture.[54]
Geometric variational problems
Further information: Willmore conjecture
In 1982, Li and Yau resolved the Willmore conjecture in the non-embedded case.[LY82] More precisely, they established that, given any smooth immersion of a closed surface in the 3-sphere which fails to be an embedding, the Willmore energy is bounded below by 8π. This is complemented by a 2012 result of Fernando Marques and André Neves, which says that in the alternative case of a smooth embedding of the 2-dimensional torus S1 × S1, the Willmore energy is bounded below by 2π2.[55] Together, these results comprise the full Willmore conjecture, as originally formulated by Thomas Willmore in 1965. Although their assumptions and conclusions are quite similar, the methods of Li−Yau and Marques−Neves are distinct. Nonetheless, they both rely on structurally similar minimax schemes. Marques and Neves made novel use of the Almgren–Pitts min-max theory of the area functional from geometric measure theory; Li and Yau's approach depended on their new "conformal invariant", which is a min-max quantity based on the Dirichlet energy. The main work of their article is devoted to relating their conformal invariant to other geometric quantities.
William Meeks and Yau produced some foundational results on minimal surfaces in three-dimensional manifolds, revisiting points left open by older work of Jesse Douglas and Charles Morrey.[MY82][41] Following these foundations, Meeks, Leon Simon, and Yau gave a number of fundamental results on surfaces in three-dimensional Riemannian manifolds which minimize area within their homology class.[MSY82] They were able to give a number of striking applications. For example, they showed that if M is an orientable 3-manifold such that every smooth embedding of a 2-sphere can be extended to a smooth embedding of the unit ball, then the same is true of any covering space of M. Interestingly, Meeks-Simon-Yau's paper and Hamilton's foundational paper on Ricci flow, published in the same year, have a result in common, obtained by very distinct methods: any simply-connected compact 3-dimensional Riemannian manifold with positive Ricci curvature is diffeomorphic to the 3-sphere.
Geometric rigidity theorems
In the geometry of submanifolds, both the extrinsic and intrinsic geometries are significant. These are reflected by the intrinsic Riemannian metric and the second fundamental form. Many geometers have considered the phenomena which arise from restricting these data to some form of constancy. This includes as special cases the problems of minimal surfaces, constant mean curvature, and submanifolds whose metric has constant scalar curvature.
• The archetypical example of such questions is Bernstein's problem, as completely settled in famous work of James Simons, Enrico Bombieri, Ennio De Giorgi, and Enrico Giusti in the 1960s. Their work asserts that a minimal hypersurface which is a graph over Euclidean space must be a plane in low dimensions, with counterexamples in high dimensions.[56] The key point of the proof of planarity is the non-existence of conical and non-planar stable minimal hypersurfaces of Euclidean spaces of low dimension; this was given a simple proof by Richard Schoen, Leon Simon, and Yau.[SSY75] Their technique of combining the Simons inequality with the formula for second variation of area has subsequently been used many times in the literature.[41][57]
• Given the "threshold" dimension phenomena in the standard Bernstein problem, it is a somewhat surprising fact, due to Shiu-Yuen Cheng and Yau, that there is no dimensional restriction in the Lorentzian analogue: any spacelike hypersurface of multidimensional Minkowski space which is a graph over Euclidean space and has zero mean curvature must be a plane.[CY76a] Their proof makes use of the maximum principle techniques which they had previously used to prove differential Harnack estimates.[CY75] Later they made use of similar techniques to give a new proof of the classification of complete parabolic or elliptic affine hyperspheres in affine geometry.[CY86]
• In one of his earliest papers, Yau considered the extension of the constant mean curvature condition to higher codimension, where the condition can be interpreted either as the mean curvature being parallel as a section of the normal bundle, or as the constancy of the length of the mean curvature. Under the former interpretation, he fully characterized the case of two-dimensional surfaces in Riemannian space forms, and found partial results under the (weaker) second interpretation.[Y74] Some of his results were independently found by Bang-Yen Chen.[58]
• Extending Philip Hartman and Louis Nirenberg's earlier work on intrinsically flat hypersurfaces of Euclidean space, Cheng and Yau considered hypersurfaces of space forms which have constant scalar curvature.[59] The key tool in their analysis was an extension of Hermann Weyl's differential identity used in the solution of the Weyl isometric embedding problem.[CY77b]
Outside of the setting of submanifold rigidity problems, Yau was able to adapt Jürgen Moser's method of proving Caccioppoli inequalities, thereby proving new rigidity results for functions on complete Riemannian manifolds. A particularly famous result of his says that a subharmonic function cannot be both positive and Lp integrable unless it is constant.[Y76][47][60] Similarly, on a complete Kähler manifold, a holomorphic function cannot be Lp integrable unless it is constant.[Y76]
Minkowski problem and Monge–Ampère equation
The Minkowski problem of classical differential geometry can be viewed as the problem of prescribing Gaussian curvature. In the 1950s, Louis Nirenberg and Aleksei Pogorelov resolved the problem for two-dimensional surfaces, making use of recent progress on the Monge–Ampère equation for two-dimensional domains. By the 1970s, higher-dimensional understanding of the Monge–Ampère equation was still lacking. In 1976, Shiu-Yuen Cheng and Yau resolved the Minkowski problem in general dimensions via the method of continuity, making use of fully geometric estimates instead of the theory of the Monge–Ampère equation.[CY76b][61]
As a consequence of their resolution of the Minkowski problem, Cheng and Yau were able to make progress on the understanding of the Monge–Ampère equation.[CY77a] The key observation is that the Legendre transform of a solution of the Monge–Ampère equation has its graph's Gaussian curvature prescribed by a simple formula depending on the "right-hand side" of the Monge–Ampère equation. As a consequence, they were able to prove the general solvability of the Dirichlet problem for the Monge–Ampère equation, which at the time had been a major open question except for two-dimensional domains.[61]
Cheng and Yau's papers followed some ideas presented in 1971 by Pogorelov, although his publicly available works (at the time of Cheng and Yau's work) had lacked some significant detail.[62] Pogorelov also published a more detailed version of his original ideas, and the resolutions of the problems are commonly attributed to both Cheng–Yau and Pogorelov.[63][61] The approaches of Cheng−Yau and Pogorelov are no longer commonly seen in the literature on the Monge–Ampère equation, as other authors, notably Luis Caffarelli, Nirenberg, and Joel Spruck, have developed direct techniques which yield more powerful results, and which do not require the auxiliary use of the Minkowski problem.[63]
Affine spheres are naturally described by solutions of certain Monge–Ampère equations, so that their full understanding is significantly more complicated than that of Euclidean spheres, the latter not being based on partial differential equations. In the parabolic case, affine spheres were completely classified as paraboloids by successive work of Konrad Jörgens, Eugenio Calabi, and Pogorelov. The elliptic affine spheres were identified as ellipsoids by Calabi. The hyperbolic affine spheres exhibit more complicated phenomena. Cheng and Yau proved that they are asymptotic to convex cones, and conversely that every (uniformly) convex cone corresponds in such a way to some hyperbolic affine sphere.[CY86] They were also able to provide new proofs of the previous classifications of Calabi and Jörgens–Calabi–Pogorelov.[61][64]
Mirror symmetry
Further information: Mirror symmetry (string theory) and SYZ conjecture
A Calabi–Yau manifold is a compact Kähler manifold which is Ricci-flat; as a special case of Yau's verification of the Calabi conjecture, such manifolds are known to exist.[Y78a] Mirror symmetry, which is a proposal developed by theoretical physicists dating from the late 1980s, postulates that Calabi−Yau manifolds of complex dimension three can be grouped into pairs which share certain characteristics, such as Euler and Hodge numbers. Based on this conjectural picture, the physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes proposed a formula of enumerative geometry which encodes the number of rational curves of any fixed degree in a general quintic hypersurface of four-dimensional complex projective space. Bong Lian, Kefeng Liu, and Yau gave a rigorous proof that this formula holds.[LLY97] A year earlier, Alexander Givental had published a proof of the mirror formulas; according to Lian, Liu, and Yau, the details of his proof were only successfully filled in following their own publication.[25] The proofs of Givental and Lian–Liu–Yau have some overlap but are distinct approaches to the problem, and each have since been given textbook expositions.[65][66]
The works of Givental and of Lian−Liu−Yau confirm a prediction made by the more fundamental mirror symmetry conjecture of how three-dimensional Calabi−Yau manifolds can be paired off. However, their works do not logically depend on the conjecture itself, and so have no immediate bearing on its validity. With Andrew Strominger and Eric Zaslow, Yau proposed a geometric picture of how mirror symmetry might be systematically understood and proved to be true.[SYZ96] Their idea is that a Calabi−Yau manifold with complex dimension three should be foliated by special Lagrangian tori, which are certain types of three-dimensional minimal submanifolds of the six-dimensional Riemannian manifold underlying the Calabi−Yau structure. Mirror manifolds would then be characterized, in terms of this conjectural structure, by having dual foliations. The Strominger−Yau−Zaslow (SYZ) proposal has been modified and developed in various ways since 1996. The conceptual picture that it provides has had a significant influence in the study of mirror symmetry, and research on its various aspects is currently an active field. It can be contrasted with the alternative homological mirror symmetry proposal by Maxim Kontsevich. The viewpoint of the SYZ conjecture is on geometric phenomena in Calabi–Yau spaces, while Kontsevich's conjecture abstracts the problem to deal with purely algebraic structures and category theory.[32][39][65][66]
Comparison geometry
In one of Yau's earliest papers, written with Blaine Lawson, a number of fundamental results were found on the topology of closed Riemannian manifolds with nonpositive curvature.[LY72] Their flat torus theorem characterizes the existence of a flat and totally geodesic immersed torus in terms of the algebra of the fundamental group. The splitting theorem says that the splitting of the fundamental group as a maximally noncommutative direct product implies the isometric splitting of the manifold itself. Similar results were obtained at the same time by Detlef Gromoll and Joseph Wolf.[67][68] Their results have been extended to the broader context of isometric group actions on metric spaces of nonpositive curvature.[69]
Jeff Cheeger and Yau studied the heat kernel on a Riemannian manifold. They established the special case of Riemannian metrics for which geodesic spheres have constant mean curvature, which they proved to be characterized by radial symmetry of the heat kernel.[CY81] Specializing to rotationally symmetric metrics, they used the exponential map to transplant the heat kernel to a geodesic ball on a general Riemannian manifold. Under the assumption that the symmetric "model" space under-estimates the Ricci curvature of the manifold itself, they carried out a direct calculation showing that the resulting function is a subsolution of the heat equation. As a consequence, they obtained a lower estimate of the heat kernel on a general Riemannian manifold in terms of lower bounds on its Ricci curvature.[70][71] In the special case of nonnegative Ricci curvature, Peter Li and Yau were able to use their gradient estimates to amplify and improve the Cheeger−Yau estimate.[LY86][47]
A well-known result of Yau's, obtained independently by Calabi, shows that any noncompact Riemannian manifold of nonnegative Ricci curvature must have volume growth of at least a linear rate.[Y76][47] A second proof, using the Bishop–Gromov inequality instead of function theory, was later found by Cheeger, Mikhael Gromov, and Michael Taylor.
Spectral geometry
Further information: spectral geometry
Given a smooth compact Riemannian manifold, with or without boundary, spectral geometry studies the eigenvalues of the Laplace–Beltrami operator, which in the case that the manifold has a boundary is coupled with a choice of boundary condition, usually Dirichlet or Neumann conditions. Paul Yang and Yau showed that in the case of a closed two-dimensional manifold, the first eigenvalue is bounded above by an explicit formula depending only on the genus and volume of the manifold.[YY80][41] Earlier, Yau had modified Jeff Cheeger's analysis of the Cheeger constant so as to be able to estimate the first eigenvalue from below in terms of geometric data.[Y75a][72]
In the 1910s, Hermann Weyl showed that, in the case of Dirichlet boundary conditions on a smooth and bounded open subset of the plane, the eigenvalues have an asymptotic behavior which is dictated entirely by the area contained in the region. His result is known as Weyl's law. In 1960, George Pólya conjectured that the Weyl law actually gives control of each individual eigenvalue, and not only of their asymptotic distribution. Li and Yau proved a weakened version of Pólya's conjecture, obtaining control of the averages of the eigenvalues by the expression in the Weyl law.[LY83][73]
In 1980, Li and Yau identified a number of new inequalities for Laplace–Beltrami eigenvalues, all based on the maximum principle and the differential Harnack estimates as pioneered five years earlier by Yau and Cheng−Yau.[LY80] Their result on lower bounds based on geometric data is particularly well-known,[74][51][47] and was the first of its kind to not require any conditional assumptions.[75] Around the same time, a similar inequality was obtained by isoperimetric methods by Mikhael Gromov, although his result is weaker than Li and Yau's.[70] In collaboration with Isadore Singer, Bun Wong, and Shing-Toung Yau, Yau used the Li–Yau methodology to establish a gradient estimate for the quotient of the first two eigenfunctions.[S+85] Analogously to Yau's integration of gradient estimates to find Harnack inequalities, they were able to integrate their gradient estimate to obtain control of the fundamental gap, which is the difference between the first two eigenvalues. The work of Singer–Wong–Yau–Yau initiated a series of works by various authors in which new estimates on the fundamental gap were found and improved.[76]
In 1982, Yau identified fourteen problems of interest in spectral geometry, including the above Pólya conjecture.[Y82b] A particular conjecture of Yau's, on the control of the size of level sets of eigenfunctions by the value of the corresponding eigenvalue, was resolved by Alexander Logunov and Eugenia Malinnikova, who were awarded the 2017 Clay Research Award in part for their work.[77]
Discrete and computational geometry
Xianfeng Gu and Yau considered the numerical computation of conformal maps between two-dimensional manifolds (presented as discretized meshes), and in particular the computation of uniformizing maps as predicted by the uniformization theorem. In the case of genus-zero surfaces, a map is conformal if and only if it is harmonic, and so Gu and Yau are able to compute conformal maps by direct minimization of a discretized Dirichlet energy.[GY02] In the case of higher genus, the uniformizing maps are computed from their gradients, as determined from the Hodge theory of closed and harmonic 1-forms.[GY02] The main work is thus to identify numerically effective discretizations of the classical theory. Their approach is sufficiently flexible to deal with general surfaces with boundary.[GY03][78] With Tony Chan, Paul Thompson, and Yalin Wang, Gu and Yau applied their work to the problem of matching two brain surfaces, which is an important issue in medical imaging. In the most-relevant genus-zero case, conformal maps are only well-defined up to the action of the Möbius group. By further optimizing a Dirichlet-type energy which measures the mismatch of brain landmarks such as the central sulcus, they obtained mappings which are well-defined by such neurological features.[G+04]
In the field of graph theory, Fan Chung and Yau extensively developed analogues of notions and results from Riemannian geometry. These results on differential Harnack inequalities, Sobolev inequalities, and heat kernel analysis, found partly in collaboration with Ronald Graham and Alexander Grigor'yan, were later written into textbook form as the last few chapters of her well-known book "Spectral Graph Theory".[79] Later, they introduced a Green's function as defined for graphs, amounting to a pseudo-inverse of the graph Laplacian.[CY00] Their work is naturally applicable to the study of hitting times for random walks and related topics.[80][81]
In the interest of finding general graph-theoretic contexts for their results, Chung and Yau introduced a notion of Ricci-flatness of a graph.[79] A more flexible notion of Ricci curvature, dealing with Markov chains on metric spaces, was later introduced by Yann Ollivier. Yong Lin, Linyuan Lu, and Yau developed some of the basic theory of Ollivier's definition in the special context of graph theory, considering for instance the Ricci curvature of Erdös–Rényi random graphs.[LLY11] Lin and Yau also considered the curvature–dimension inequalities introduced earlier by Dominique Bakry and Michel Émery, relating it and Ollivier's curvature to Chung–Yau's notion of Ricci-flatness.[LY10] They were further able to prove general lower bounds on Bakry–Émery and Ollivier's curvatures in the case of locally finite graphs.[82]
Honors and awards
Yau has received honorary professorships from many Chinese universities, including Hunan Normal University, Peking University, Nankai University, and Tsinghua University. He has honorary degrees from many international universities, including Harvard University, Chinese University of Hong Kong, and University of Waterloo. He is a foreign member of the National Academies of Sciences of China, India, and Russia.
His awards include:
• 1975–1976, Sloan Fellow.
• 1981, Oswald Veblen Prize in Geometry.
• 1981, John J. Carty Award for the Advancement of Science, United States National Academy of Sciences.[83]
• 1982, Fields Medal, for "his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations."
• 1982, elected to the American Academy of Arts and Sciences
• 1982, Guggenheim Fellowship.
• 1984–1985, MacArthur Fellow.
• 1991, Humboldt Research Award, Alexander von Humboldt Foundation, Germany.
• 1993, elected to the United States National Academy of Sciences
• 1994, Crafoord Prize.[84]
• 1997, United States National Medal of Science.
• 2003, China International Scientific and Technological Cooperation Award, for "his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers."
• 2010, Wolf Prize in Mathematics, for "his work in geometric analysis and mathematical physics".[85]
• 2018, Marcel Grossmann Awards, "for the proof of the positivity of total mass in the theory of general relativity and perfecting as well the concept of quasi-local mass, for his proof of the Calabi conjecture, for his continuous inspiring role in the study of black holes physics."[86]
• 2023, Shaw Prize in Mathematical Sciences.[87]
Major publications
Research articles. Yau is the author of over five hundred articles. The following, among the most cited, are surveyed above:
LY72.
Lawson, H. Blaine Jr.; Yau, Shing Tung (1972). "Compact manifolds of nonpositive curvature". Journal of Differential Geometry. 7 (1–2): 211–228. doi:10.4310/jdg/1214430828. MR 0334083. Zbl 0266.53035.
Y74.
Yau, Shing Tung (1974). "Submanifolds with constant mean curvature. I". American Journal of Mathematics. 96 (2): 346–366. doi:10.2307/2373638. JSTOR 2373638. MR 0370443. Zbl 0304.53041.
CY75.
Cheng, S. Y.; Yau, S. T. (1975). "Differential equations on Riemannian manifolds and their geometric applications". Communications on Pure and Applied Mathematics. 28 (3): 333–354. doi:10.1002/cpa.3160280303. MR 0385749. Zbl 0312.53031.
SSY75.
Schoen, R.; Simon, L.; Yau, S. T. (1975). "Curvature estimates for minimal hypersurfaces". Acta Mathematica. 134 (3–4): 275–288. doi:10.1007/BF02392104. MR 0423263. Zbl 0323.53039.
Y75a.
Yau, Shing Tung (1975). "Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold". Annales Scientifiques de l'École Normale Supérieure. Quatrième Série. 8 (4): 487–507. doi:10.24033/asens.1299. MR 0397619. Zbl 0325.53039.
Y75b.
Yau, Shing Tung (1975). "Harmonic functions on complete Riemannian manifolds". Communications on Pure and Applied Mathematics. 28 (2): 201–228. doi:10.1002/cpa.3160280203. MR 0431040. Zbl 0291.31002.
CY76a.
Cheng, Shiu Yuen; Yau, Shing Tung (1976). "Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces". Annals of Mathematics. Second Series. 104 (3): 407–419. doi:10.2307/1970963. JSTOR 1970963. MR 0431061. Zbl 0352.53021.
CY76b.
Cheng, Shiu Yuen; Yau, Shing Tung (1976). "On the regularity of the solution of the n-dimensional Minkowski problem". Communications on Pure and Applied Mathematics. 29 (5): 495–516. doi:10.1002/cpa.3160290504. MR 0423267. Zbl 0363.53030.
SY76.
Schoen, Richard; Yau, Shing Tung (1976). "Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature". Commentarii Mathematici Helvetici. 51 (3): 333–341. doi:10.1007/BF02568161. MR 0438388. S2CID 120845708. Zbl 0361.53040.
Y76.
Yau, Shing Tung (1976). "Some function-theoretic properties of complete Riemannian manifold and their applications to geometry". Indiana University Mathematics Journal. 25 (7): 659–670. doi:10.1512/iumj.1976.25.25051. MR 0417452. Zbl 0335.53041. (Erratum: doi:10.1512/iumj.1982.31.31044)
CY77a.
Cheng, Shiu Yuen; Yau, Shing Tung (1977). "On the regularity of the Monge–Ampère equation det(∂2u/∂xi∂xj) = F(x,u)". Communications on Pure and Applied Mathematics. 30 (1): 41–68. doi:10.1002/cpa.3160300104. MR 0437805. Zbl 0347.35019.
CY77b.
Cheng, Shiu Yuen; Yau, Shing Tung (1977). "Hypersurfaces with constant scalar curvature". Mathematische Annalen. 225 (3): 195–204. doi:10.1007/BF01425237. MR 0431043. S2CID 33626481. Zbl 0349.53041.
Y77.
Yau, Shing Tung (1977). "Calabi's conjecture and some new results in algebraic geometry". Proceedings of the National Academy of Sciences of the United States of America. 74 (5): 1798–1799. Bibcode:1977PNAS...74.1798Y. doi:10.1073/pnas.74.5.1798. MR 0451180. PMC 431004. PMID 16592394. Zbl 0355.32028.
Y78a.
Yau, Shing Tung (1978). "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I". Communications on Pure and Applied Mathematics. 31 (3): 339–411. doi:10.1002/cpa.3160310304. MR 0480350. Zbl 0369.53059.
Y78b.
Yau, Shing Tung (1978). "A general Schwarz lemma for Kähler manifolds". American Journal of Mathematics. 100 (1): 197–203. doi:10.2307/2373880. JSTOR 2373880. MR 0486659. Zbl 0424.53040.
SY79a.
Schoen, R.; Yau, Shing Tung (1979). "Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature". Annals of Mathematics. Second Series. 110 (1): 127–142. doi:10.2307/1971247. JSTOR 1971247. MR 0541332. Zbl 0431.53051.
SY79b.
Schoen, R.; Yau, S. T. (1979). "On the structure of manifolds with positive scalar curvature". Manuscripta Mathematica. 28 (1–3): 159–183. doi:10.1007/BF01647970. MR 0535700. S2CID 121008386. Zbl 0423.53032.
SY79c.
Schoen, Richard; Yau, Shing Tung (1979). "On the proof of the positive mass conjecture in general relativity". Communications in Mathematical Physics. 65 (1): 45–76. Bibcode:1979CMaPh..65...45S. doi:10.1007/BF01940959. MR 0526976. S2CID 54217085. Zbl 0405.53045.
CY80.
Cheng, Shiu Yuen; Yau, Shing Tung (1980). "On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation". Communications on Pure and Applied Mathematics. 33 (4): 507–544. doi:10.1002/cpa.3160330404. MR 0575736. Zbl 0506.53031.
LY80.
Li, Peter; Yau, Shing Tung (1980). "Estimates of eigenvalues of a compact Riemannian manifold". In Osserman, Robert; Weinstein, Alan (eds.). Geometry of the Laplace Operator. University of Hawaii, Honolulu (March 27–30, 1979). Proceedings of Symposia in Pure Mathematics. Vol. 36. Providence, RI: American Mathematical Society. pp. 205–239. doi:10.1090/pspum/036. ISBN 9780821814390. MR 0573435. Zbl 0441.58014.
SY80.
Siu, Yum Tong; Yau, Shing Tung (1980). "Compact Kähler manifolds of positive bisectional curvature". Inventiones Mathematicae. 59 (2): 189–204. Bibcode:1980InMat..59..189S. doi:10.1007/BF01390043. MR 0577360. S2CID 120664058. Zbl 0442.53056.
YY80.
Yang, Paul C.; Yau, Shing Tung (1980). "Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 7 (1): 55–63. MR 0577325. Zbl 0446.58017.
CY81.
Cheeger, Jeff; Yau, Shing-Tung (1981). "A lower bound for the heat kernel". Communications on Pure and Applied Mathematics. 34 (4): 465–480. doi:10.1002/cpa.3160340404. MR 0615626. Zbl 0481.35003.
CLY81.
Cheng, Siu Yuen; Li, Peter; Yau, Shing-Tung (1981). "On the upper estimate of the heat kernel of a complete Riemannian manifold". American Journal of Mathematics. 103 (5): 1021–1063. doi:10.2307/2374257. JSTOR 2374257. MR 0630777. Zbl 0484.53035.
SY81.
Schoen, Richard; Yau, Shing Tung (1981). "Proof of the positive mass theorem. II". Communications in Mathematical Physics. 79 (2): 231–260. Bibcode:1981CMaPh..79..231S. doi:10.1007/BF01942062. MR 0612249. S2CID 59473203. Zbl 0494.53028.
LY82.
Li, Peter; Yau, Shing Tung (1982). "A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces". Inventiones Mathematicae. 69 (2): 269–291. Bibcode:1982InMat..69..269L. doi:10.1007/BF01399507. MR 0674407. S2CID 123019753. Zbl 0503.53042.
MSY82.
Meeks, William, III; Simon, Leon; Yau, Shing Tung (1982). "Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature". Annals of Mathematics. Second Series. 116 (3): 621–659. doi:10.2307/2007026. JSTOR 2007026. MR 0678484. Zbl 0521.53007.{{cite journal}}: CS1 maint: multiple names: authors list (link)
MY82.
Meeks, William H., III; Yau, Shing Tung (1982). "The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas–Morrey and an analytic proof of Dehn's lemma". Topology. 21 (4): 409–442. doi:10.1016/0040-9383(82)90021-0. MR 0670745. Zbl 0489.57002.{{cite journal}}: CS1 maint: multiple names: authors list (link)
LY83.
Li, Peter; Yau, Shing Tung (1983). "On the Schrödinger equation and the eigenvalue problem". Communications in Mathematical Physics. 88 (3): 309–318. Bibcode:1983CMaPh..88..309L. doi:10.1007/BF01213210. MR 0701919. S2CID 120055958. Zbl 0554.35029.
SY83.
Schoen, Richard; Yau, S. T. (1983). "The existence of a black hole due to condensation of matter". Communications in Mathematical Physics. 90 (4): 575–579. Bibcode:1983CMaPh..90..575S. doi:10.1007/BF01216187. MR 0719436. S2CID 122331620. Zbl 0541.53054.
S+85.
Singer, I. M.; Wong, Bun; Yau, Shing-Tung; Yau, Stephen S.-T. (1985). "An estimate of the gap of the first two eigenvalues in the Schrödinger operator". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 12 (2): 319–333. MR 0829055. Zbl 0603.35070.
CY86.
Cheng, Shiu Yuen; Yau, Shing-Tung (1986). "Complete affine hypersurfaces. I. The completeness of affine metrics". Communications on Pure and Applied Mathematics. 39 (6): 839–866. doi:10.1002/cpa.3160390606. MR 0859275. Zbl 0623.53002.
LY86.
Li, Peter; Yau, Shing-Tung (1986). "On the parabolic kernel of the Schrödinger operator". Acta Mathematica. 156 (3–4): 153–201. doi:10.1007/bf02399203. MR 0834612. Zbl 0611.58045.
UY86.
Uhlenbeck, K.; Yau, S.-T. (1986). "On the existence of Hermitian–Yang–Mills connections in stable vector bundles". Communications on Pure and Applied Mathematics. 39 (S): 257–293. doi:10.1002/cpa.3160390714. MR 0861491. Zbl 0615.58045. (Erratum: doi:10.1002/cpa.3160420505)
SY88.
Schoen, R.; Yau, S.-T. (1988). "Conformally flat manifolds, Kleinian groups and scalar curvature". Inventiones Mathematicae. 92 (1): 47–71. Bibcode:1988InMat..92...47S. doi:10.1007/BF01393992. MR 0931204. S2CID 59029712. Zbl 0658.53038.
G+90.
Greene, Brian R.; Shapere, Alfred; Vafa, Cumrun; Yau, Shing-Tung (1990). "Stringy cosmic strings and noncompact Calabi–Yau manifolds". Nuclear Physics B. 337 (1): 1–36. Bibcode:1990NuPhB.337....1G. doi:10.1016/0550-3213(90)90248-C. MR 1059826. Zbl 0744.53045.
TY90.
Tian, G.; Yau, Shing-Tung (1990). "Complete Kähler manifolds with zero Ricci curvature. I". Journal of the American Mathematical Society. 3 (3): 579–609. doi:10.1090/S0894-0347-1990-1040196-6. MR 1040196. Zbl 0719.53041.
TY91.
Tian, Gang; Yau, Shing-Tung (1991). "Complete Kähler manifolds with zero Ricci curvature. II". Inventiones Mathematicae. 106 (1): 27–60. Bibcode:1991InMat.106...27T. doi:10.1007/BF01243902. MR 1123371. S2CID 122638262. Zbl 0766.53053.
HY96.
Huisken, Gerhard; Yau, Shing-Tung (1996). "Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature". Inventiones Mathematicae. 124 (1–3): 281–311. Bibcode:1996InMat.124..281H. doi:10.1007/s002220050054. hdl:11858/00-001M-0000-0013-5B63-3. MR 1369419. S2CID 122669931. Zbl 0858.53071.
SYZ96.
Strominger, Andrew; Yau, Shing-Tung; Zaslow, Eric (1996). "Mirror symmetry is T-duality". Nuclear Physics B. 479 (1–2): 243–259. arXiv:hep-th/9606040. Bibcode:1996NuPhB.479..243S. doi:10.1016/0550-3213(96)00434-8. MR 1429831. S2CID 14586676. Zbl 0896.14024.
LLY97.
Lian, Bong H.; Liu, Kefeng; Yau, Shing-Tung (1997). "Mirror principle. I". Asian Journal of Mathematics. 1 (4): 729–763. arXiv:alg-geom/9712011. Bibcode:1997alg.geom.12011L. doi:10.4310/AJM.1997.v1.n4.a5. MR 1621573. Zbl 0953.14026.
CY00.
Chung, Fan; Yau, S.-T. (2000). "Discrete Green's functions". Journal of Combinatorial Theory. Series A. 91 (1–2): 191–214. doi:10.1006/jcta.2000.3094. MR 1779780. Zbl 0963.65120.
GY02.
Gu, Xianfeng; Yau, Shing-Tung (2002). "Computing conformal structures of surfaces". Communications in Information and Systems. 2 (2): 121–145. arXiv:cs/0212043. Bibcode:2002cs.......12043G. doi:10.4310/CIS.2002.v2.n2.a2. MR 1958012. Zbl 1092.14514.
GY03.
Gu, Xianfeng; Yau, Shing Tung (2003). "Global conformal surface parameterization". In Kobbelt, Leif; Schroeder, Peter; Hoppe, Hugues (eds.). Eurographics Symposium on Geometry Processing (Aachen, Germany, June 23–25, 2003). Goslar, Germany: Eurographics Association. pp. 127–137. doi:10.2312/SGP/SGP03/127-137.
G+04.
Gu, Xianfeng; Wang, Yalin; Chan, Tony F.; Thompson, Paul M.; Yau, Shing-Tung (2004). "Genus zero surface conformal mapping and its application to brain surface mapping". IEEE Transactions on Medical Imaging. 28 (8): 949–958. doi:10.1109/TMI.2004.831226. PMID 15338729.
FY08.
Fu, Ji-Xiang; Yau, Shing-Tung (2008). "The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation". Journal of Differential Geometry. 78 (3): 369–428. doi:10.4310/jdg/1207834550. MR 2396248. Zbl 1141.53036.
LY10.
Lin, Yong; Yau, Shing-Tung (2010). "Ricci curvature and eigenvalue estimate on locally finite graphs". Mathematical Research Letters. 17 (2): 343–356. doi:10.4310/MRL.2010.v17.n2.a13. MR 2644381. Zbl 1232.31003.
LLY11.
Lin, Yong; Lu, Linyuan; Yau, Shing-Tung (2011). "Ricci curvature of graphs". Tohoku Mathematical Journal. Second Series. 63 (4): 605–627. doi:10.2748/tmj/1325886283. MR 2872958. Zbl 1237.05204.
Survey articles and publications of collected works.
Y82a.
Yau, Shing Tung (1982). "Survey on partial differential equations in differential geometry". In Yau, Shing-Tung (ed.). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ: Princeton University Press. pp. 3–71. doi:10.1515/9781400881918-002. ISBN 9781400881918. MR 0645729. Zbl 0478.53001.
Y82b.
Yau, Shing Tung (1982). "Problem section". In Yau, Shing-Tung (ed.). Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ: Princeton University Press. pp. 669–706. doi:10.1515/9781400881918-035. ISBN 9781400881918. MR 0645762. Zbl 0479.53001.
Y87.
Yau, Shing-Tung (1987). "Nonlinear analysis in geometry". L'Enseignement Mathématique. Revue Internationale. 2e Série. 33 (1–2): 109–158. doi:10.5169/seals-87888. MR 0896385. Zbl 0631.53002.
Y93.
Yau, Shing-Tung (1993). "Open problems in geometry". In Greene, Robert; Yau, S. T. (eds.). Differential Geometry: Partial Differential Equations on Manifolds. American Mathematical Society Summer Institute on Differential Geometry (University of California, Los Angeles, July 9–27, 1990). Proceedings of Symposia in Pure Mathematics. Vol. 54. Providence, RI: American Mathematical Society. pp. 1–28. doi:10.1090/pspum/054.1. ISBN 9780821814949. MR 1216573. Zbl 0801.53001.
Y00.
Yau, S.-T. (2000). "Review of geometry and analysis". Asian Journal of Mathematics. 4 (1): 235–278. doi:10.4310/AJM.2000.v4.n1.a16. MR 1803723. Zbl 1031.53004.
Y06.
Yau, Shing-Tung (2006). "Perspectives on geometric analysis". In Yau, Shing-Tung (ed.). Essays in geometry in memory of S.S. Chern. Surveys in Differential Geometry. Vol. 10. Somerville, MA: International Press. pp. 275–379. doi:10.4310/SDG.2005.v10.n1.a8. MR 2408227. Zbl 1138.53004.
Y14a.
Ji, Lizhen; Li, Peter; Liu, Kefeng; Schoen, Richard, eds. (2014a). Selected expository works of Shing-Tung Yau with commentary. Vol. I. Advanced Lectures in Mathematics. Vol. 28. Somerville, MA: International Press. ISBN 978-1-57146-293-0. MR 3307244. Zbl 1401.01045.
Y14b.
Ji, Lizhen; Li, Peter; Liu, Kefeng; Schoen, Richard, eds. (2014b). Selected expository works of Shing-Tung Yau with commentary. Vol. II. Advanced Lectures in Mathematics. Vol. 29. Somerville, MA: International Press. ISBN 978-1-57146-294-7. MR 3307245. Zbl 1401.01046.
Y19a.
Cao, Huai-Dong; Li, Jun; Schoen, Richard, eds. (2019a). Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 1: Metric geometry and minimal submanifolds. Somerville, MA: International Press. ISBN 978-1-57146-376-0. Zbl 1412.01037.
Y19b.
Cao, Huai-Dong; Li, Jun; Schoen, Richard, eds. (2019b). Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 2: Metric geometry and harmonic functions. Somerville, MA: International Press. ISBN 978-1-57146-377-7. Zbl 1412.01038.
Y19c.
Cao, Huai-Dong; Li, Jun; Schoen, Richard, eds. (2019c). Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 3: Eigenvalues and general relativity. Somerville, MA: International Press. ISBN 978-1-57146-378-4. Zbl 1412.01039.
Y19d.
Cao, Huai-Dong; Li, Jun; Schoen, Richard, eds. (2019d). Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 4: Kähler geometry I. Somerville, MA: International Press. ISBN 978-1-57146-379-1. Zbl 1412.01040.
Y19e.
Cao, Huai-Dong; Li, Jun; Schoen, Richard, eds. (2019e). Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 5: Kähler geometry II. Somerville, MA: International Press. ISBN 978-1-57146-380-7. Zbl 1412.01041.
Textbooks and technical monographs.
SY94.
Schoen, R.; Yau, S.-T. (1994). Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology. Vol. 1. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang, Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S. Y. Cheng. Preface translated from the Chinese by Kaising Tso. Cambridge, MA: International Press. ISBN 1-57146-012-8. MR 1333601. Zbl 0830.53001.
SY97.
Schoen, R.; Yau, S. T. (1997). Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology. Vol. 2. Cambridge, MA: International Press. ISBN 1-57146-002-0. MR 1474501. Zbl 0886.53004.
SY98.
Salaff, Stephen; Yau, Shing-Tung (1998). Ordinary differential equations (Second ed.). Cambridge, MA: International Press. ISBN 1-57146-065-9. MR 1691427. Zbl 1089.34500.
GY08.
Gu, Xianfeng David; Yau, Shing-Tung (2008). Computational conformal geometry. Advanced Lectures in Mathematics. Vol. 3. Somerville, MA: International Press. ISBN 978-1-57146-171-1. MR 2439718.
Popular books.
YN10.
Yau, Shing-Tung; Nadis, Steve (2010). The shape of inner space. String theory and the geometry of the universe's hidden dimensions. New York: Basic Books. ISBN 978-0-465-02023-2. MR 2722198. Zbl 1235.00025.
NY13.
Nadis, Steve; Yau, Shing-Tung (2013). A history in sum. 150 years of mathematics at Harvard (1825–1975). Cambridge, MA: Harvard University Press. ISBN 978-0-674-72500-3. MR 3100544. Zbl 1290.01005.
YN19.
Yau, Shing-Tung; Nadis, Steve (2019). The shape of a life. One mathematician's search for the universe's hidden geometry. New Haven, CT: Yale University Press. ISBN 978-0-300-23590-6. MR 3930611. Zbl 1435.32001.
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87. Shaw Prize 2023
External links
• Center of Mathematical Sciences at Zhejiang University: commentary by various mathematicians on Yau
• Discover Magazine Interview, June 2010 issue
• Interview (11 pages long in Traditional Chinese)
• Yau's autobiographical account (mostly English, some Chinese)
• O'Connor, John J.; Robertson, Edmund F., "Shing-Tung Yau", MacTutor History of Mathematics Archive, University of St Andrews
• Shing-Tung Yau at the Mathematics Genealogy Project
• Plugging A Math Gap
• UC Irvine courting Yau with a $2.5 million professorship
• International Conference Celebrating Shing Tung Yau's Birthday 8/27/2008-9/1/2008 Harvard University
Fields Medalists
• 1936 Ahlfors
• Douglas
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Wikipedia
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Quadrilateralized spherical cube
In mapmaking, a quadrilateralized spherical cube, or quad sphere for short, is an equal-area polyhedral map projection and discrete global grid scheme for data collected on a spherical surface (either that of the Earth or the celestial sphere). It was first proposed in 1975 by Chan and O'Neill for the Naval Environmental Prediction Research Facility.[2] This scheme is also often called the COBE sky cube,[3] because it was designed to hold data from the Cosmic Background Explorer (COBE) project.[4]
Elements
The quad sphere has two principal characteristic features. The first is that the mapping consists of projecting the sphere onto the faces of an inscribed cube using a curvilinear projection that preserves area. The sphere is divided into six equal regions, which correspond to the faces of the cube. The vertices of the cube correspond to the cartesian coordinates defined by |x|=|y|=|z| on a sphere centred at the origin. For an Earth projection, the cube is usually oriented with one face normal to the North Pole and one face centered on the Greenwich meridian (although any definition of pole and meridian could be used). The faces of the cube are divided into a grid of square bins, where the number of bins along each edge is a power of 2, selected to produce the desired bin size. Thus the number of bins on each face is 22N, where N is the binning depth, for a total of 6 × 22N. For example, a binning depth of 10 gives 1024 × 1024 bins on each face or 6291456 (6 × 220) in all, each bin covering an area of 23.6 square arcminutes (2.00 microsteradians).
The second key feature is that the bins are numbered serially, rather than being rastered as for an image. The total number of bits required for the bin numbers at level N is 2N + 3, where the three most significant bits are used for the face numbers and the remaining bits are used to number the bins within each face. The faces are numbered from 0 to 5: 0 for the north face, 1 through 4 for the equatorial faces (1 being on the meridian), and 5 for the south. Thus at a binning depth of 10, face 0 has bin numbers 0–1,048,575, face 1 has numbers 1,048,576–2,097,151, and so on. Within each face the bins are numbered serially from one corner (the convention is to start at the "lower left") to the opposite corner, ordered in such a way that each pair of bits corresponds to a level of bin resolution. This ordering is in effect a two-dimensional binary tree, which is referred to as the quad-tree. The conversion between bin numbers and coordinates is straightforward. If four-byte integers are used for the bin numbers the maximum practical depth, which uses 31 of the 32 bits, results in a bin size of 0.0922 square arcminutes (7.80 nanosteradians).
In principle, the mapping and numbering schemes are separable: the map projection onto the cube could be used with another bin-numbering scheme, and the numbering scheme itself could be used with any arrangement of bins susceptible to partitioning into a set of square arrays. Used together, they make a flexible and efficient system for storing map data.
Advantages
The quad sphere projection does not produce singularities at the poles or elsewhere, as do some other equal-area mapping schemes. Distortion is moderate over the entire sphere, so that at no point are shapes altered beyond recognition.
Related projections
There are some related projections:
• rHEALPix: is a cubic configuration in the HEALPix framework (of 2003), elaborated in 2016.[5]
• S2 projection was created at Google (published in 2016 with a first pre-release in 2019) for the purpose of defining a discrete global grid scheme. It is similar to the quad sphere but is not equal-area.[6][7]
See also
• List of map projections
• Cube mapping
• Geodesic grid
References
1. "Quadrilateralized Spherical Cube — PROJ 9.2.1 documentation". proj.org. Retrieved 2023-06-10.
2. Chan, F.K.; O'Neill, E. M. (1975). Feasibility Study of a Quadrilateralized Spherical Cube Earth Data Base (CSC - Computer Sciences Corporation, EPRF Technical Report 2-75) (Technical report). Monterey, California: Environmental Prediction Research Facility.
3. "COBE Quadrilateralized Spherical Cube".
4. Max Tegmark. "What is the best way to pixelize a sphere?".
5. Gibb, R G (April 2016). "The rHEALPix Discrete Global Grid System". IOP Conference Series: Earth and Environmental Science. 34: 012012. doi:10.1088/1755-1315/34/1/012012. ISSN 1755-1307. S2CID 64092160.
6. "S2 — PROJ 8.2.1 documentation". proj.org. Retrieved 2022-02-19.
7. "S2 Geometry". S2Geometry. Retrieved 2022-02-19.
• O'Neill, E. M. (1976). Extended Studies of a Quadrilateralized Spherical Cube Earth Data Base (PDF) (Technical report). Monterey, California: Environmental Prediction Research Facility. Archived (PDF) from the original on May 7, 2019.
• Fred Patt (Feb 18, 1993). "Comments on Draft WCS Standard". Newsgroup: sci.astro.fits. Usenet: [email protected]. Retrieved Jul 8, 2021.
Map projection
• History
• List
• Portal
By surface
Cylindrical
Mercator-conformal
• Gauss–Krüger
• Transverse Mercator
• Oblique Mercator
Equal-area
• Balthasart
• Behrmann
• Gall–Peters
• Hobo–Dyer
• Lambert
• Smyth equal-surface
• Trystan Edwards
• Cassini
• Central
• Equirectangular
• Gall stereographic
• Gall isographic
• Miller
• Space-oblique Mercator
• Web Mercator
Pseudocylindrical
Equal-area
• Collignon
• Eckert II
• Eckert IV
• Eckert VI
• Equal Earth
• Goode homolosine
• Mollweide
• Sinusoidal
• Tobler hyperelliptical
• Kavrayskiy VII
• Wagner VI
• Winkel I and II
Conical
• Albers
• Equidistant
• Lambert conformal
Pseudoconical
• Bonne
• Bottomley
• Polyconic
• American
• Chinese
• Werner
Azimuthal
(planar)
General perspective
• Gnomonic
• Orthographic
• Stereographic
• Equidistant
• Lambert equal-area
Pseudoazimuthal
• Aitoff
• Hammer
• Wiechel
• Winkel tripel
By metric
Conformal
• Adams hemisphere-in-a-square
• Gauss–Krüger
• Guyou hemisphere-in-a-square
• Lambert conformal conic
• Mercator
• Peirce quincuncial
• Stereographic
• Transverse Mercator
Equal-area
Bonne
• Sinusoidal
• Werner
Bottomley
• Sinusoidal
• Werner
Cylindrical
• Balthasart
• Behrmann
• Gall–Peters
• Hobo–Dyer
• Lambert cylindrical equal-area
• Smyth equal-surface
• Trystan Edwards
Tobler hyperelliptical
• Collignon
• Mollweide
• Albers
• Briesemeister
• Eckert II
• Eckert IV
• Eckert VI
• Equal Earth
• Goode homolosine
• Hammer
• Lambert azimuthal equal-area
• Quadrilateralized spherical cube
• Strebe 1995
Equidistant in
some aspect
• Conic
• Equirectangular
• Sinusoidal
• Two-point
• Werner
Gnomonic
• Gnomonic
Loxodromic
• Loximuthal
• Mercator
Retroazimuthal
(Mecca or Qibla)
• Craig
• Hammer
• Littrow
By construction
Compromise
• Chamberlin trimetric
• Kavrayskiy VII
• Miller cylindrical
• Natural Earth
• Robinson
• Van der Grinten
• Wagner VI
• Winkel tripel
Hybrid
• Goode homolosine
• HEALPix
Perspective
Planar
• Gnomonic
• Orthographic
• Stereographic
• Central cylindrical
Polyhedral
• AuthaGraph
• Cahill Butterfly
• Cahill–Keyes M-shape
• Dymaxion
• ISEA
• Quadrilateralized spherical cube
• Waterman butterfly
See also
• Interruption (map projection)
• Latitude
• Longitude
• Tissot's indicatrix
• Map projection of the tri-axial ellipsoid
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Space diagonal
In geometry, a space diagonal (also interior diagonal or body diagonal) of a polyhedron is a line connecting two vertices that are not on the same face. Space diagonals contrast with face diagonals, which connect vertices on the same face (but not on the same edge) as each other.[1]
For example, a pyramid has no space diagonals, while a cube (shown at right) or more generally a parallelepiped has four space diagonals.
Axial diagonal
An axial diagonal is a space diagonal that passes through the center of a polyhedron.
For example, in a cube with edge length a, all four space diagonals are axial diagonals, of common length $a{\sqrt {3}}.$ More generally, a cuboid with edge lengths a, b, and c has all four space diagonals axial, with common length ${\sqrt {a^{2}+b^{2}+c^{2}}}.$
A regular octahedron has 3 axial diagonals, of length $a{\sqrt {2}}$, with edge length a.
A regular icosahedron has 6 axial diagonals of length $a{\sqrt {2+\varphi }}$, where $\varphi $ is the golden ratio $(1+{\sqrt {5}})/2$.[2]
Space diagonals of magic cubes
Main article: Magic cube
A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.
See also
• Distance
• Face diagonal
• Magic cube classes
• Hypotenuse
• Spacetime interval
References
1. William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.116
2. Sutton, Daud (2002), Platonic & Archimedean Solids, Wooden Books, Bloomsbury Publishing USA, p. 55, ISBN 9780802713865.
• John R. Hendricks, The Pan-3-Agonal Magic Cube, Journal of Recreational Mathematics 5:1:1972, pp 51–54. First published mention of pan-3-agonals
• Hendricks, J. R., Magic Squares to Tesseracts by Computer, 1998, 0-9684700-0-9, page 49
• Heinz & Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0, pages 99,165
• Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 173, 1994.
External links
• Weisstein, Eric W. "Space Diagonals". MathWorld.
• de Winkel Magic Encyclopedia
• Heinz - Basic cube parts
• John Hendricks Hypercubes
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Divine Proportions: Rational Trigonometry to Universal Geometry
Divine Proportions: Rational Trigonometry to Universal Geometry is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocates replacing the usual basic quantities of trigonometry, Euclidean distance and angle measure, by squared distance and the square of the sine of the angle, respectively. This is logically equivalent to the standard development (as the replacement quantities can be expressed in terms of the standard ones and vice versa). The author claims his approach holds some advantages, such as avoiding the need for irrational numbers.
The book was "essentially self-published"[1] by Wildberger through his publishing company Wild Egg. The formulas and theorems in the book are regarded as correct mathematics but the claims about practical or pedagogical superiority are primarily promoted by Wildberger himself and have received mixed reviews.
Overview
The main idea of Divine Proportions is to replace distances by the squared Euclidean distance, which Wildberger calls the quadrance, and to replace angle measures by the squares of their sines, which Wildberger calls the spread between two lines. Divine Proportions defines both of these concepts directly from the Cartesian coordinates of points that determine a line segment or a pair of crossing lines. Defined in this way, they are rational functions of those coordinates, and can be calculated directly without the need to take the square roots or inverse trigonometric functions required when computing distances or angle measures.[1]
For Wildberger, a finitist, this replacement has the purported advantage of avoiding the concepts of limits and actual infinity used in defining the real numbers, which Wildberger claims to be unfounded.[2][1] It also allows analogous concepts to be extended directly from the rational numbers to other number systems such as finite fields using the same formulas for quadrance and spread.[1] Additionally, this method avoids the ambiguity of the two supplementary angles formed by a pair of lines, as both angles have the same spread. This system is claimed to be more intuitive, and to extend more easily from two to three dimensions.[3] However, in exchange for these benefits, one loses the additivity of distances and angles: for instance, if a line segment is divided in two, its length is the sum of the lengths of the two pieces, but combining the quadrances of the pieces is more complicated and requires square roots.[1]
Organization and topics
Divine Proportions is divided into four parts. Part I presents an overview of the use of quadrance and spread to replace distance and angle, and makes the argument for their advantages. Part II formalizes the claims made in part I, and proves them rigorously.[1] Rather than defining lines as infinite sets of points, they are defined by their homogeneous coordinates, which may be used in formulas for testing the incidence of points and lines. Like the sine, the cosine and tangent are replaced with rational equivalents, called the "cross" and "twist", and Divine Proportions develops various analogues of trigonometric identities involving these quantities,[3] including versions of the Pythagorean theorem, law of sines and law of cosines.[4]
Part III develops the geometry of triangles and conic sections using the tools developed in the two previous parts.[1] Well known results such as Heron's formula for calculating the area of a triangle from its side lengths, or the inscribed angle theorem in the form that the angles subtended by a chord of a circle from other points on the circle are equal, are reformulated in terms of quadrance and spread, and thereby generalized to arbitrary fields of numbers.[3][5] Finally, Part IV considers practical applications in physics and surveying, and develops extensions to higher-dimensional Euclidean space and to polar coordinates.[1]
Audience
Divine Proportions does not assume much in the way of mathematical background in its readers, but its many long formulas, frequent consideration of finite fields, and (after part I) emphasis on mathematical rigor are likely to be obstacles to a popular mathematics audience. Instead, it is mainly written for mathematics teachers and researchers. However, it may also be readable by mathematics students, and contains exercises making it possible to use as the basis for a mathematics course.[1][6]
Critical reception
The feature of the book that was most positively received by reviewers was its work extending results in distance and angle geometry to finite fields. Reviewer Laura Wiswell found this work impressive, and was charmed by the result that the smallest finite field containing a regular pentagon is $\mathbb {F} _{19}$.[1] Michael Henle calls the extension of triangle and conic section geometry to finite fields, in part III of the book, "an elegant theory of great generality",[4] and William Barker also writes approvingly of this aspect of the book, calling it "particularly novel" and possibly opening up new research directions.[6]
Wiswell raises the question of how many of the detailed results presented without attribution in this work are actually novel.[1] In this light, Michael Henle notes that the use of squared Euclidean distance "has often been found convenient elsewhere";[4] for instance it is used in distance geometry, least squares statistics, and convex optimization. James Franklin points out that for spaces of three or more dimensions, modeled conventionally using linear algebra, the use of spread by Divine Proportions is not very different from standard methods involving dot products in place of trigonometric functions.[5]
An advantage of Wildberger's methods noted by Henle is that, because they involve only simple algebra, the proofs are both easy to follow and easy for a computer to verify. However, he suggests that the book's claims of greater simplicity in its overall theory rest on a false comparison in which quadrance and spread are weighed not against the corresponding classical concepts of distances, angles, and sines, but the much wider set of tools from classical trigonometry. He also points out that, to a student with a scientific calculator, formulas that avoid square roots and trigonometric functions are a non-issue,[4] and Barker adds that the new formulas often involve a greater number of individual calculation steps.[6] Although multiple reviewers felt that a reduction in the amount of time needed to teach students trigonometry would be very welcome,[3][5][7] Paul Campbell is skeptical that these methods would actually speed learning.[7] Gerry Leversha keeps an open mind, writing that "It will be interesting to see some of the textbooks aimed at school pupils [that Wildberger] has promised to produce, and ... controlled experiments involving student guinea pigs."[3] As of 2020, however, these textbooks and experiments have not been published.
Wiswell is unconvinced by the claim that conventional geometry has foundational flaws that these methods avoid.[1] While agreeing with Wiswell, Barker points out that there may be other mathematicians who share Wildberger's philosophical suspicions of the infinite, and that this work should be of great interest to them.[6]
A final issue raised by multiple reviewers is inertia: supposing for the sake of argument that these methods are better, are they enough better to make worthwhile the large individual effort of re-learning geometry and trigonometry in these terms, and the institutional effort of re-working the school curriculum to use them in place of classical geometry and trigonometry? Henle, Barker, and Leversha conclude that the book has not made its case for this,[3][4][6] but Sandra Arlinghaus sees this work as an opportunity for fields such as her mathematical geography "that have relatively little invested in traditional institutional rigidity" to demonstrate the promise of such a replacement.[8]
See also
• Perles configuration, a finite set of points and lines in the Euclidean plane that cannot be represented with rational coordinates
References
1. Wiswell, Laura (June 2007), "Review of Divine Proportions", Proceedings of the Edinburgh Mathematical Society, 50 (2): 509–510, doi:10.1017/S0013091507215020, ProQuest 228292466
2. Gefter, Amanda (2013), "Mind-bending mathematics: Why infinity has to go", New Scientist, 219 (2930): 32–35, doi:10.1016/s0262-4079(13)62043-6
3. Leversha, Gerry (March 2008), "Review of Divine Proportions", The Mathematical Gazette, 92 (523): 184–186, doi:10.1017/S0025557200182944, JSTOR 27821758, S2CID 125430473
4. Henle, Michael (December 2007), "Review of Divine Proportions", The American Mathematical Monthly, 114 (10): 933–937, JSTOR 27642383
5. Franklin, James (June 2006), "Review of Divine Proportions" (PDF), The Mathematical Intelligencer, 28 (3): 73–74, doi:10.1007/bf02986892, S2CID 121754449
6. Barker, William (July 2008), "Review of Divine Proportions", MAA Reviews, Mathematical Association of America
7. Campbell, Paul J. (February 2007), "Review of Divine Proportions", Mathematics Magazine, 80 (1): 84–85, doi:10.1080/0025570X.2007.11953460, JSTOR 27643001, S2CID 218543379
8. Arlinghaus, Sandra L. (June 2006), "Review of Divine Proportions", Solstice: An Electronic Journal of Geography and Mathematics, 17 (1), hdl:2027.42/60314
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Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
For less elementary aspects of the subject, see Polynomial ring.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
Etymology
The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century.[1]
Notation and terminology
The x occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.
A polynomial P in the indeterminate x is commonly denoted either as P or as P(x). Formally, the name of the polynomial is P, not P(x), but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let P(x) be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.
The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If a denotes a number, a variable, another polynomial, or, more generally, any expression, then P(a) denotes, by convention, the result of substituting a for x in P. Thus, the polynomial P defines the function
$a\mapsto P(a),$
which is the polynomial function associated to P. Frequently, when using this notation, one supposes that a is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any ring). In particular, if a is a polynomial then P(a) is also a polynomial.
More specifically, when a is the indeterminate x, then the image of x by this function is the polynomial P itself (substituting x for x does not change anything). In other words,
$P(x)=P,$
which justifies formally the existence of two notations for the same polynomial.
Definition
A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power. The constants are generally numbers, but may be any expression that do not involve the indeterminates, and represent mathematical objects that can be added and multiplied. Two polynomial expressions are considered as defining the same polynomial if they may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication. For example $(x-1)(x-2)$ and $x^{2}-3x+2$ are two polynomial expressions that represent the same polynomial; so, one has the equality $(x-1)(x-2)=x^{2}-3x+2$.
A polynomial in a single indeterminate x can always be written (or rewritten) in the form
$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0},$
where $a_{0},\ldots ,a_{n}$ are constants that are called the coefficients of the polynomial, and $x$ is the indeterminate.[2] The word "indeterminate" means that $x$ represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function.
This can be expressed more concisely by using summation notation:
$\sum _{k=0}^{n}a_{k}x^{k}$
That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number – called the coefficient of the term[lower-alpha 1] – and a finite number of indeterminates, raised to non-negative integer powers.
Classification
Further information: Degree of a polynomial
The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any term with nonzero coefficient.[3] Because x = x1, the degree of an indeterminate without a written exponent is one.
A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial.[lower-alpha 2] The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial 0 (which has no terms at all) is generally treated as not defined (but see below).[4]
For example:
$-5x^{2}y$
is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3.
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
$\underbrace {_{\,}3x^{2}} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {1} \end{smallmatrix}}\underbrace {-_{\,}5x} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {2} \end{smallmatrix}}\underbrace {+_{\,}4} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {3} \end{smallmatrix}}.$
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial, or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials.[3] For higher degrees, the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term 2x in x2 + 2x + 1 is a linear term in a quadratic polynomial.
The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).[5] The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, f(x) = 0, is the x-axis.
In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined.[lower-alpha 3] For example, x3y2 + 7x2y3 − 3x5 is homogeneous of degree 5. For more details, see Homogeneous polynomial.
The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial 3x2 - 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.[6]
Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0.[7] Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial,[lower-alpha 4] a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial.
A real polynomial is a polynomial with real coefficients. When it is used to define a function, the domain is not so restricted. However, a real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients.
A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. A polynomial with two indeterminates is called a bivariate polynomial.[2] These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials (which may result from the subtraction of non-constant polynomials), although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is also common to say simply "polynomials in x, y, and z", listing the indeterminates allowed.
Operations
Addition and subtraction
Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms.[7][8] For example, if
$P=3x^{2}-2x+5xy-2$ and $Q=-3x^{2}+3x+4y^{2}+8$
then the sum
$P+Q=3x^{2}-2x+5xy-2-3x^{2}+3x+4y^{2}+8$
can be reordered and regrouped as
$P+Q=(3x^{2}-3x^{2})+(-2x+3x)+5xy+4y^{2}+(8-2)$
and then simplified to
$P+Q=x+5xy+4y^{2}+6.$
When polynomials are added together, the result is another polynomial.[9]
Subtraction of polynomials is similar.
Multiplication
Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.[7] For example, if
${\begin{aligned}\color {Red}P&\color {Red}{=2x+3y+5}\\\color {Blue}Q&\color {Blue}{=2x+5y+xy+1}\end{aligned}}$
then
${\begin{array}{rccrcrcrcr}{\color {Red}{P}}{\color {Blue}{Q}}&{=}&&({\color {Red}{2x}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{3y}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{5}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{5}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{5}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{5}}\cdot {\color {Blue}{1}})\end{array}}$
Carrying out the multiplication in each term produces
${\begin{array}{rccrcrcrcr}PQ&=&&4x^{2}&+&10xy&+&2x^{2}y&+&2x\\&&+&6xy&+&15y^{2}&+&3xy^{2}&+&3y\\&&+&10x&+&25y&+&5xy&+&5.\end{array}}$
Combining similar terms yields
${\begin{array}{rcccrcrcrcr}PQ&=&&4x^{2}&+&(10xy+6xy+5xy)&+&2x^{2}y&+&(2x+10x)\\&&+&15y^{2}&+&3xy^{2}&+&(3y+25y)&+&5\end{array}}$
which can be simplified to
$PQ=4x^{2}+21xy+2x^{2}y+12x+15y^{2}+3xy^{2}+28y+5.$
As in the example, the product of polynomials is always a polynomial.[9][4]
Composition
Given a polynomial $f$ of a single variable and another polynomial g of any number of variables, the composition $f\circ g$ is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.[4] For example, if $f(x)=x^{2}+2x$ and $g(x)=3x+2$ then
$(f\circ g)(x)=f(g(x))=(3x+2)^{2}+2(3x+2).$
A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.[10]
Division
The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context.[11] This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer.[12][13] For example, the fraction 1/(x2 + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x.
For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers.[lower-alpha 5] This notion of the division a(x)/b(x) results in two polynomials, a quotient q(x) and a remainder r(x), such that a = b q + r and degree(r) < degree(b). The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.[14]
When the denominator b(x) is monic and linear, that is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation a(c).[13] In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.[15]
Factoring
All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree.[16] For example, the factored form of
$5x^{3}-5$
is
$5(x-1)\left(x^{2}+x+1\right)$
over the integers and the reals, and
$5(x-1)\left(x+{\frac {1+i{\sqrt {3}}}{2}}\right)\left(x+{\frac {1-i{\sqrt {3}}}{2}}\right)$
over the complex numbers.
The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.
Calculus
Main article: Calculus with polynomials
Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions.
The derivative of the polynomial
$P=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}$
with respect to x is the polynomial
$na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\dots +2a_{2}x+a_{1}=\sum _{i=1}^{n}ia_{i}x^{i-1}.$
Similarly, the general antiderivative (or indefinite integral) of $P$ is
${\frac {a_{n}x^{n+1}}{n+1}}+{\frac {a_{n-1}x^{n}}{n}}+\dots +{\frac {a_{2}x^{3}}{3}}+{\frac {a_{1}x^{2}}{2}}+a_{0}x+c=c+\sum _{i=0}^{n}{\frac {a_{i}x^{i+1}}{i+1}}$
where c is an arbitrary constant. For example, antiderivatives of x2 + 1 have the form 1/3x3 + x + c.
For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number p, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient kak understood to mean the sum of k copies of ak. For example, over the integers modulo p, the derivative of the polynomial xp + x is the polynomial 1.[17]
Polynomial functions
See also: Ring of polynomial functions
A polynomial function is a function that can be defined by evaluating a polynomial. More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial
$a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}$
that evaluates to $f(x)$ for all x in the domain of f (here, n is a non-negative integer and a0, a1, a2, ..., an are constant coefficients). Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals.
For example, the function f, defined by
$f(x)=x^{3}-x,$
is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in
$f(x,y)=2x^{3}+4x^{2}y+xy^{5}+y^{2}-7.$
According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression $\left({\sqrt {1-x^{2}}}\right)^{2},$ which takes the same values as the polynomial $1-x^{2}$ on the interval $[-1,1]$, and thus both expressions define the same polynomial function on this interval.
Every polynomial function is continuous, smooth, and entire.
The evaluation of a polynomial is the computation of the corresponding polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions.
For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method, which consists of rewriting the polynomial as
$(((((a_{n}x+a_{n-1})x+a_{n-2})x+\dotsb +a_{3})x+a_{2})x+a_{1})x+a_{0}.$
Graphs
• Polynomial of degree 0:
f(x) = 2
• Polynomial of degree 1:
f(x) = 2x + 1
• Polynomial of degree 2:
f(x) = x2 − x − 2
= (x + 1)(x − 2)
• Polynomial of degree 3:
f(x) = x3/4 + 3x2/4 − 3x/2 − 2
= 1/4 (x + 4)(x + 1)(x − 2)
• Polynomial of degree 4:
f(x) = 1/14 (x + 4)(x + 1)(x − 1)(x − 3)
+ 0.5
• Polynomial of degree 5:
f(x) = 1/20 (x + 4)(x + 2)(x + 1)(x − 1)
(x − 3) + 2
• Polynomial of degree 6:
f(x) = 1/100 (x6 − 2x 5 − 26x4 + 28x3
+ 145x2 − 26x − 80)
• Polynomial of degree 7:
f(x) = (x − 3)(x − 2)(x − 1)(x)(x + 1)(x + 2)
(x + 3)
A polynomial function in one real variable can be represented by a graph.
• The graph of the zero polynomial
f(x) = 0
is the x-axis.
• The graph of a degree 0 polynomial
f(x) = a0, where a0 ≠ 0,
is a horizontal line with y-intercept a0
• The graph of a degree 1 polynomial (or linear function)
f(x) = a0 + a1x, where a1 ≠ 0,
is an oblique line with y-intercept a0 and slope a1.
• The graph of a degree 2 polynomial
f(x) = a0 + a1x + a2x2, where a2 ≠ 0
is a parabola.
• The graph of a degree 3 polynomial
f(x) = a0 + a1x + a2x2 + a3x3, where a3 ≠ 0
is a cubic curve.
• The graph of any polynomial with degree 2 or greater
f(x) = a0 + a1x + a2x2 + ⋯ + anxn, where an ≠ 0 and n ≥ 2
is a continuous non-linear curve.
A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction (one branch for positive x and one for negative x).
Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.
Equations
Main article: Algebraic equation
A polynomial equation, also called an algebraic equation, is an equation of the form[18]
$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}=0.$
For example,
$3x^{2}+4x-5=0$
is a polynomial equation.
When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like (x + y)(x − y) = x2 − y2, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.
In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals. However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.
The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra.
Solving equations
Main article: Algebraic equation
See also: Root-finding of polynomials and Properties of polynomial roots
A root of a nonzero univariate polynomial P is a value a of x such that P(a) = 0. In other words, a root of P is a solution of the polynomial equation P(x) = 0 or a zero of the polynomial function defined by P. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.
A number a is a root of a polynomial P if and only if the linear polynomial x − a divides P, that is if there is another polynomial Q such that P = (x − a) Q. It may happen that a power (greater than 1) of x − a divides P; in this case, a is a multiple root of P, and otherwise a is a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (x − a)m divides P, which is called the multiplicity of a as a root of P. The number of roots of a nonzero polynomial P, counted with their respective multiplicities, cannot exceed the degree of P,[19] and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra). The coefficients of a polynomial and its roots are related by Vieta's formulas.
Some polynomials, such as x2 + 1, do not have any roots among the real numbers. If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.
There may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of 2x − 1 = 0 is 1/2. Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expressions; for example, the golden ratio $(1+{\sqrt {5}})/2$ is the unique positive solution of $x^{2}-x-1=0.$ In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see cubic equation and quartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel–Ruffini theorem). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and group theory, two important branches of modern algebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).
When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving it is to compute numerical approximations of the solutions.[20] There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm).
For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called zeros instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry. For a set of polynomial equations with several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions. See System of polynomial equations.
The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.
A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general algorithm for solving them, or even for deciding whether the set of solutions is empty (see Hilbert's tenth problem). Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as Fermat's Last Theorem.
Polynomial expressions
Polynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.
Trigonometric polynomials
Main article: Trigonometric polynomial
A trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers.[21] The coefficients may be taken as real numbers, for real-valued functions.
If sin(nx) and cos(nx) are expanded in terms of sin(x) and cos(x), a trigonometric polynomial becomes a polynomial in the two variables sin(x) and cos(x) (using List of trigonometric identities#Multiple-angle formulae). Conversely, every polynomial in sin(x) and cos(x) may be converted, with Product-to-sum identities, into a linear combination of functions sin(nx) and cos(nx). This equivalence explains why linear combinations are called polynomials.
For complex coefficients, there is no difference between such a function and a finite Fourier series.
Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are also used in the discrete Fourier transform.
Matrix polynomials
A matrix polynomial is a polynomial with square matrices as variables.[22] Given an ordinary, scalar-valued polynomial
$P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},$
this polynomial evaluated at a matrix A is
$P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},$
where I is the identity matrix.[23]
A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn(R).
Exponential polynomials
A bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example P(x, ex), may be called an exponential polynomial.
Related concepts
Rational functions
Main article: Rational function
A rational fraction is the quotient (algebraic fraction) of two polynomials. Any algebraic expression that can be rewritten as a rational fraction is a rational function.
While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not zero.
The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.
Laurent polynomials
Main article: Laurent polynomial
Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.
Power series
Main article: Formal power series
Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like irrational numbers cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formal power series also generalize polynomials, but the multiplication of two power series may not converge.
Polynomial ring
Main article: Polynomial ring
A polynomial f over a commutative ring R is a polynomial all of whose coefficients belong to R. It is straightforward to verify that the polynomials in a given set of indeterminates over R form a commutative ring, called the polynomial ring in these indeterminates, denoted $R[x]$ in the univariate case and $R[x_{1},\ldots ,x_{n}]$ in the multivariate case.
One has
$R[x_{1},\ldots ,x_{n}]=\left(R[x_{1},\ldots ,x_{n-1}]\right)[x_{n}].$
So, most of the theory of the multivariate case can be reduced to an iterated univariate case.
The map from R to R[x] sending r to itself considered as a constant polynomial is an injective ring homomorphism, by which R is viewed as a subring of R[x]. In particular, R[x] is an algebra over R.
One can think of the ring R[x] as arising from R by adding one new element x to R, and extending in a minimal way to a ring in which x satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is xr = rx). To do this, one must add all powers of x and their linear combinations as well.
Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring R[x] over the real numbers by factoring out the ideal of multiples of the polynomial x2 + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).
If R is commutative, then one can associate with every polynomial P in R[x] a polynomial function f with domain and range equal to R. (More generally, one can take domain and range to be any same unital associative algebra over R.) One obtains the value f(r) by substitution of the value r for the symbol x in P. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for x.
Divisibility
Main articles: Polynomial greatest common divisor and Factorization of polynomials
If R is an integral domain and f and g are polynomials in R[x], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[x] such that f q = g. If $a\in R,$ then a is a root of f if and only $x-a$ divides f. In this case, the quotient can be computed using the polynomial long division.[24][25]
If F is a field and f and g are polynomials in F[x] with g ≠ 0, then there exist unique polynomials q and r in F[x] with
$f=q\,g+r$
and such that the degree of r is smaller than the degree of g (using the convention that the polynomial 0 has a negative degree). The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain.
Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials. In the case of coefficients in a ring, "non-constant" must be replaced by "non-constant or non-unit" (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Eisenstein's criterion can also be used in some cases to determine irreducibility.
Applications
Positional notation
Main article: Positional notation
In modern positional numbers systems, such as the decimal system, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the radix or base, in this case, 4 × 101 + 5 × 100. As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number 1 × 52 + 3 × 51 + 2 × 50 = 42. This representation is unique. Let b be a positive integer greater than 1. Then every positive integer a can be expressed uniquely in the form
$a=r_{m}b^{m}+r_{m-1}b^{m-1}+\dotsb +r_{1}b+r_{0},$
where m is a nonnegative integer and the r's are integers such that
0 < rm < b and 0 ≤ ri < b for i = 0, 1, . . . , m − 1.[26]
Interpolation and approximation
See also: Polynomial interpolation, Orthogonal polynomials, B-spline, and spline interpolation
The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include polynomial interpolation and the use of splines.[27]
Other applications
Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.
The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in computational complexity theory the phrase polynomial time means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input.
History
Main articles: Cubic function § History, Quartic function § History, and Abel–Ruffini theorem § History
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, c. 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.
History of the notation
Main article: History of mathematical notation
The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.[28]
See also
• List of polynomial topics
Notes
1. See "polynomial" and "binomial", Compact Oxford English Dictionary
2. Weisstein, Eric W. "Polynomial". mathworld.wolfram.com. Retrieved 2020-08-28.
3. "Polynomials | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-28.
4. Barbeau 2003, pp. 1–2
5. Weisstein, Eric W. "Zero Polynomial". MathWorld.
6. Edwards 1995, p. 78
7. Edwards, Harold M. (1995). Linear Algebra. Springer. p. 47. ISBN 978-0-8176-3731-6.
8. Salomon, David (2006). Coding for Data and Computer Communications. Springer. p. 459. ISBN 978-0-387-23804-3.
9. Introduction to Algebra. Yale University Press. 1965. p. 621. Any two such polynomials can be added, subtracted, or multiplied. Furthermore , the result in each case is another polynomial
10. Kriete, Hartje (1998-05-20). Progress in Holomorphic Dynamics. CRC Press. p. 159. ISBN 978-0-582-32388-9. This class of endomorphisms is closed under composition,
11. Marecek, Lynn; Mathis, Andrea Honeycutt (6 May 2020). Intermediate Algebra 2e. OpenStax. §7.1.
12. Haylock, Derek; Cockburn, Anne D. (2008-10-14). Understanding Mathematics for Young Children: A Guide for Foundation Stage and Lower Primary Teachers. SAGE. p. 49. ISBN 978-1-4462-0497-9. We find that the set of integers is not closed under this operation of division.
13. Marecek & Mathis 2020, §5.4]
14. Selby, Peter H.; Slavin, Steve (1991). Practical Algebra: A Self-Teaching Guide (2nd ed.). Wiley. ISBN 978-0-471-53012-1.
15. Weisstein, Eric W. "Ruffini's Rule". mathworld.wolfram.com. Retrieved 2020-07-25.
16. Barbeau 2003, pp. 80–2
17. Barbeau 2003, pp. 64–5
18. Proskuryakov, I.V. (1994). "Algebraic equation". In Hazewinkel, Michiel (ed.). Encyclopaedia of Mathematics. Vol. 1. Springer. ISBN 978-1-55608-010-4.
19. Leung, Kam-tim; et al. (1992). Polynomials and Equations. Hong Kong University Press. p. 134. ISBN 9789622092716.
20. McNamee, J.M. (2007). Numerical Methods for Roots of Polynomials, Part 1. Elsevier. ISBN 978-0-08-048947-6.
21. Powell, Michael J. D. (1981). Approximation Theory and Methods. Cambridge University Press. ISBN 978-0-521-29514-7.
22. Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics. Vol. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-681-8. Zbl 1170.15300.
23. Horn & Johnson 1990, p. 36.
24. Irving, Ronald S. (2004). Integers, Polynomials, and Rings: A Course in Algebra. Springer. p. 129. ISBN 978-0-387-20172-6.
25. Jackson, Terrence H. (1995). From Polynomials to Sums of Squares. CRC Press. p. 143. ISBN 978-0-7503-0329-3.
26. McCoy 1968, p. 75
27. de Villiers, Johann (2012). Mathematics of Approximation. Springer. ISBN 9789491216503.
28. Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Saunders. ISBN 0-03-029558-0.
1. The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo some prime number p.
2. This terminology dates from the time when the distinction was not clear between a polynomial and the function that it defines: a constant term and a constant polynomial define constant functions.
3. In fact, as a homogeneous function, it is homogeneous of every degree.
4. Some authors use "monomial" to mean "monic monomial". See Knapp, Anthony W. (2007). Advanced Algebra: Along with a Companion Volume Basic Algebra. Springer. p. 457. ISBN 978-0-8176-4522-9.
5. This paragraph assumes that the polynomials have coefficients in a field.
References
• Barbeau, E.J. (2003). Polynomials. Springer. ISBN 978-0-387-40627-5.
• Bronstein, Manuel; et al., eds. (2006). Solving Polynomial Equations: Foundations, Algorithms, and Applications. Springer. ISBN 978-3-540-27357-8.
• Cahen, Paul-Jean; Chabert, Jean-Luc (1997). Integer-Valued Polynomials. American Mathematical Society. ISBN 978-0-8218-0388-2.
• Horn, Roger A.; Johnson, Charles R. (1990). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6..
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556. This classical book covers most of the content of this article.
• Leung, Kam-tim; et al. (1992). Polynomials and Equations. Hong Kong University Press. ISBN 9789622092716.
• Mayr, K. (1937). "Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen". Monatshefte für Mathematik und Physik. 45: 280–313. doi:10.1007/BF01707992. S2CID 197662587.
• McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225
• Prasolov, Victor V. (2005). Polynomials. Springer. ISBN 978-3-642-04012-2.
• Sethuraman, B.A. (1997). "Polynomials". Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra Via Geometric Constructibility. Springer. ISBN 978-0-387-94848-5.
• Toth, Gabor (2021). "Polynomial Expressions". Elements of Mathematics. Undergraduate Texts in Mathematics. pp. 263–318. doi:10.1007/978-3-030-75051-0_6. ISBN 978-3-030-75050-3.
• Umemura, H. (2012) [1984]. "Resolution of algebraic equations by theta constants". In Mumford, David (ed.). Tata Lectures on Theta II: Jacobian theta functions and differential equations. Springer. pp. 261–. ISBN 978-0-8176-4578-6.
• von Lindemann, F. (1884). "Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen". Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen. 1884: 245–8.
• von Lindemann, F. (1892). "Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen. II". Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen. 1892: 245–8.
External links
Wikimedia Commons has media related to Polynomials.
Look up polynomial in Wiktionary, the free dictionary.
• Markushevich, A.I. (2001) [1994], "Polynomial", Encyclopedia of Mathematics, EMS Press
• "Euler's Investigations on the Roots of Equations". Archived from the original on September 24, 2012.
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
• Univariate
• Bivariate
• Multivariate
• Monomial
• Binomial
• Trinomial
• Irreducible
• Square-free
• Homogeneous
• Quasi-homogeneous
Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
Authority control: National
• France
• BnF data
• Israel
• United States
• Japan
• Czech Republic
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Quadrant (plane geometry)
The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes.
Not to be confused with Circle quadrant.
These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the (x; y) coordinates are I (+; +), II (−; +), III (−; −), and IV (+; −). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant.
Mnemonic
In the above graphic, the words in quotation marks are a mnemonic for remembering which three trigonometric functions (sine, cosine and tangent) are positive in each quadrant. The expression reads "All Science Teachers Crazy" and proceeding counterclockwise from the upper right quadrant, we see that "All" functions are positive in quadrant I, "Science" (for sine) is positive in quadrant II, "Teachers" (for tangent) is positive in quadrant III, and "Crazy" (for cosine) is positive in quadrant IV. There are several variants of this mnemonic.
See also
• Orthant
• Octant (solid geometry)
External links
Wikimedia Commons has media related to Quadrants.
• Weisstein, Eric W. "Quadrant". MathWorld.
• Quadrant at PlanetMath.
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Circular sector
A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector.[1] In the diagram, θ is the central angle, $r$ the radius of the circle, and $L$ is the arc length of the minor sector.
Not to be confused with circular section.
The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.[2]
Types
A sector with the central angle of 180° is called a half-disk and is bounded by a diameter and a semicircle. Sectors with other central angles are sometimes given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. Confusingly, the arc of a quadrant (a circular arc) can also be termed a quadrant.
Compass
Traditionally wind directions on the compass rose are given as one of the 8 octants (N, NE, E, SE, S, SW, W, NW) because that is more precise than merely giving one of the 4 quadrants, and the wind vane typically does not have enough accuracy to allow more precise indication.
The name of the instrument "octant" comes from the fact that it is based on 1/8th of the circle. Most commonly, octants are seen on the compass rose.
Area
See also: Circular arc § Sector area
The total area of a circle is πr2. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2π (because the area of the sector is directly proportional to its angle, and 2π is the angle for the whole circle, in radians):
$A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}$
The area of a sector in terms of L can be obtained by multiplying the total area πr2 by the ratio of L to the total perimeter 2πr.
$A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}$
Another approach is to consider this area as the result of the following integral:
$A=\int _{0}^{\theta }\int _{0}^{r}dS=\int _{0}^{\theta }\int _{0}^{r}{\tilde {r}}\,d{\tilde {r}}\,d{\tilde {\theta }}=\int _{0}^{\theta }{\frac {1}{2}}r^{2}\,d{\tilde {\theta }}={\frac {r^{2}\theta }{2}}$
Converting the central angle into degrees gives[3]
$A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}$
Perimeter
The length of the perimeter of a sector is the sum of the arc length and the two radii:
$P=L+2r=\theta r+2r=r(\theta +2)$
where θ is in radians.
Arc length
The formula for the length of an arc is:[4]
$L=r\theta $
where L represents the arc length, r represents the radius of the circle and θ represents the angle in radians made by the arc at the centre of the circle.[5]
If the value of angle is given in degrees, then we can also use the following formula by:[3]
$L=2\pi r{\frac {\theta }{360}}$
Chord length
The length of a chord formed with the extremal points of the arc is given by
$C=2R\sin {\frac {\theta }{2}}$
where C represents the chord length, R represents the radius of the circle, and θ represents the angular width of the sector in radians.
See also
• Circular segment – the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
• Conic section
• Earth quadrant
• Sector of (mathematics)
References
1. Dewan, Rajesh K. (2016). Saraswati Mathematics. New Delhi: New Saraswati House India Pvt Ltd. p. 234. ISBN 978-8173358371.
2. Achatz, Thomas; Anderson, John G. (2005). Technical shop mathematics. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376. ISBN 978-0831130862. OCLC 56559272.
3. Uppal, Shveta (2019). Mathematics: Textbook for class X. New Delhi: National Council of Educational Research and Training. pp. 226, 227. ISBN 978-81-7450-634-4. OCLC 1145113954.
4. Larson, Ron; Edwards, Bruce H. (2002). Calculus I with Precalculus (3rd ed.). Boston, MA.: Brooks/Cole. p. 570. ISBN 978-0-8400-6833-0. OCLC 706621772.
5. Wicks, Alan (2004). Mathematics Standard Level for the International Baccalaureate : a text for the new syllabus. West Conshohocken, PA: Infinity Publishing.com. p. 79. ISBN 0-7414-2141-0. OCLC 58869667.
Sources
• Gerard, L. J. V., The Elements of Geometry, in Eight Books; or, First Step in Applied Logic (London, Longmans, Green, Reader and Dyer, 1874), p. 285.
• Legendre, A. M., Elements of Geometry and Trigonometry, Charles Davies, ed. (New York: A. S. Barnes & Co., 1858), p. 119.
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Quadratic-linear algebra
In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski (2005, p.101). An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – [X, Y] = 0.
References
• Polishchuk, Alexander; Positselski, Leonid (2005), Quadratic algebras, University Lecture Series, vol. 37, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3834-1, MR 2177131
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Quadratic Gauss sum
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.
Definition
For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is defined as
$g(a;p)=\sum _{n=0}^{p-1}\zeta _{p}^{an^{2}},$
where $\zeta _{p}$ is a primitive pth root of unity, for example $\zeta _{p}=\exp(2\pi i/p)$. Equivalently,
$g(a;p)=\sum _{n=0}^{p-1}{\big (}1+\left({\tfrac {n}{p}}\right){\big )}\,\zeta _{p}^{an}.$
For a divisible by p the expression $\zeta _{p}^{an^{2}}$ evaluates to $1$. Hence, we have
$g(a;p)=p.$
For a not divisible by p, this expression reduces to
$g(a;p)=\sum _{n=0}^{p-1}\left({\tfrac {n}{p}}\right)\,\zeta _{p}^{an}=G(a,\left({\tfrac {\cdot }{p}}\right)),$
where
$G(a,\chi )=\sum _{n=0}^{p-1}\chi (n)\,\zeta _{p}^{an}$
is the Gauss sum defined for any character χ modulo p.
Properties
• The value of the Gauss sum is an algebraic integer in the pth cyclotomic field $\mathbb {Q} (\zeta _{p})$.
• The evaluation of the Gauss sum for an integer a not divisible by a prime p > 2 can be reduced to the case a = 1:
$g(a;p)=\left({\tfrac {a}{p}}\right)g(1;p).$
• The exact value of the Gauss sum for a = 1 is given by the formula:
$g(1;p)=\sum _{n=0}^{p-1}e^{\frac {2\pi in^{2}}{p}}={\begin{cases}{\sqrt {p}}&{\text{if}}\ p\equiv 1{\pmod {4}},\\i{\sqrt {p}}&{\text{if}}\ p\equiv 3{\pmod {4}}.\end{cases}}$
Remark
In fact, the identity
$g(1;p)^{2}=\left({\tfrac {-1}{p}}\right)p$
was easy to prove and led to one of Gauss's proofs of quadratic reciprocity. However, the determination of the sign of the Gauss sum turned out to be considerably more difficult: Gauss could only establish it after several years' work. Later, Dirichlet, Kronecker, Schur and other mathematicians found different proofs.
Generalized quadratic Gauss sums
Let a, b, c be natural numbers. The generalized quadratic Gauss sum G(a, b, c) is defined by
$G(a,b,c)=\sum _{n=0}^{c-1}e^{2\pi i{\frac {an^{2}+bn}{c}}}$.
The classical quadratic Gauss sum is the sum g(a, p) = G(a, 0, p).
Properties
• The Gauss sum G(a,b,c) depends only on the residue class of a and b modulo c.
• Gauss sums are multiplicative, i.e. given natural numbers a, b, c, d with gcd(c, d) = 1 one has
$G(a,b,cd)=G(ac,b,d)G(ad,b,c).$
This is a direct consequence of the Chinese remainder theorem.
• One has G(a, b, c) = 0 if gcd(a, c) > 1 except if gcd(a,c) divides b in which case one has
$G(a,b,c)=\gcd(a,c)\cdot G\left({\frac {a}{\gcd(a,c)}},{\frac {b}{\gcd(a,c)}},{\frac {c}{\gcd(a,c)}}\right)$.
Thus in the evaluation of quadratic Gauss sums one may always assume gcd(a, c) = 1.
• Let a, b, c be integers with ac ≠ 0 and ac + b even. One has the following analogue of the quadratic reciprocity law for (even more general) Gauss sums[1]
$\sum _{n=0}^{|c|-1}e^{\pi i{\frac {an^{2}+bn}{c}}}=\left|{\frac {c}{a}}\right|^{\frac {1}{2}}e^{\pi i{\frac {|ac|-b^{2}}{4ac}}}\sum _{n=0}^{|a|-1}e^{-\pi i{\frac {cn^{2}+bn}{a}}}$.
• Define
$\varepsilon _{m}={\begin{cases}1&{\text{if}}\ m\equiv 1{\pmod {4}}\\i&{\text{if}}\ m\equiv 3{\pmod {4}}\end{cases}}$
for every odd integer m. The values of Gauss sums with b = 0 and gcd(a, c) = 1 are explicitly given by
$G(a,c)=G(a,0,c)={\begin{cases}0&{\text{if}}\ c\equiv 2{\pmod {4}}\\\varepsilon _{c}{\sqrt {c}}\left({\dfrac {a}{c}}\right)&{\text{if}}\ c\equiv 1{\pmod {2}}\\(1+i)\varepsilon _{a}^{-1}{\sqrt {c}}\left({\dfrac {c}{a}}\right)&{\text{if}}\ c\equiv 0{\pmod {4}}.\end{cases}}$
Here (a/c) is the Jacobi symbol. This is the famous formula of Carl Friedrich Gauss.
• For b > 0 the Gauss sums can easily be computed by completing the square in most cases. This fails however in some cases (for example, c even and b odd), which can be computed relatively easy by other means. For example, if c is odd and gcd(a, c) = 1 one has
$G(a,b,c)=\varepsilon _{c}{\sqrt {c}}\cdot \left({\frac {a}{c}}\right)e^{-2\pi i{\frac {\psi (a)b^{2}}{c}}},$
where ψ(a) is some number with 4ψ(a)a ≡ 1 (mod c). As another example, if 4 divides c and b is odd and as always gcd(a, c) = 1 then G(a, b, c) = 0. This can, for example, be proved as follows: because of the multiplicative property of Gauss sums we only have to show that G(a, b, 2n) = 0 if n > 1 and a, b are odd with gcd(a, c) = 1. If b is odd then an2 + bn is even for all 0 ≤ n < c − 1. By Hensel's lemma, for every q, the equation an2 + bn + q = 0 has at most two solutions in $\mathbb {Z} $/2n$\mathbb {Z} $. Because of a counting argument an2 + bn runs through all even residue classes modulo c exactly two times. The geometric sum formula then shows that G(a, b, 2n) = 0.
• If c is an odd square-free integer and gcd(a, c) = 1, then
$G(a,0,c)=\sum _{n=0}^{c-1}\left({\frac {n}{c}}\right)e^{\frac {2\pi ian}{c}}.$
If c is not squarefree then the right side vanishes while the left side does not. Often the right sum is also called a quadratic Gauss sum.
• Another useful formula
$G\left(n,p^{k}\right)=p\cdot G\left(n,p^{k-2}\right)$
holds for k ≥ 2 and an odd prime number p, and for k ≥ 4 and p = 2.
See also
• Gauss sum
• Gaussian period
• Kummer sum
• Landsberg–Schaar relation
References
1. Theorem 1.2.2 in B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums, John Wiley and Sons, (1998).
• Ireland; Rosen (1990). A Classical Introduction to Modern Number Theory. Springer-Verlag. ISBN 0-387-97329-X.
• Berndt, Bruce C.; Evans, Ronald J.; Williams, Kenneth S. (1998). Gauss and Jacobi Sums. Wiley and Sons. ISBN 0-471-12807-4.
• Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic number theory. American Mathematical Society. ISBN 0-8218-3633-1.
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Quadratic Lie algebra
A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R).
Lie groups and Lie algebras
Classical groups
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
Simple Lie groups
Classical
• An
• Bn
• Cn
• Dn
Exceptional
• G2
• F4
• E6
• E7
• E8
Other Lie groups
• Circle
• Lorentz
• Poincaré
• Conformal group
• Diffeomorphism
• Loop
• Euclidean
Lie algebras
• Lie group–Lie algebra correspondence
• Exponential map
• Adjoint representation
• Killing form
• Index
• Simple Lie algebra
• Loop algebra
• Affine Lie algebra
Semisimple Lie algebra
• Dynkin diagrams
• Cartan subalgebra
• Root system
• Weyl group
• Real form
• Complexification
• Split Lie algebra
• Compact Lie algebra
Representation theory
• Lie group representation
• Lie algebra representation
• Representation theory of semisimple Lie algebras
• Representations of classical Lie groups
• Theorem of the highest weight
• Borel–Weil–Bott theorem
Lie groups in physics
• Particle physics and representation theory
• Lorentz group representations
• Poincaré group representations
• Galilean group representations
Scientists
• Sophus Lie
• Henri Poincaré
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Claude Chevalley
• Harish-Chandra
• Armand Borel
• Glossary
• Table of Lie groups
Definition
A quadratic Lie algebra is a Lie algebra (g,[.,.]) together with a non-degenerate symmetric bilinear form $(.,.)\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} $ that is invariant under the adjoint action, i.e.
([X,Y],Z)+(Y,[X,Z])=0
where X,Y,Z are elements of the Lie algebra g. A localization/ generalization is the concept of Courant algebroid where the vector space g is replaced by (sections of) a vector bundle.
Examples
As a first example, consider Rn with zero-bracket and standard inner product
$((x_{1},\dots ,x_{n}),(y_{1},\dots ,y_{n})):=\sum _{j}x_{j}y_{j}$.
Since the bracket is trivial the invariance is trivially fulfilled.
As a more elaborate example consider so(3), i.e. R3 with base X,Y,Z, standard inner product, and Lie bracket
$[X,Y]=Z,\quad [Y,Z]=X,\quad [Z,X]=Y$.
Straightforward computation shows that the inner product is indeed preserved. A generalization is the following.
Semisimple Lie algebras
A big group of examples fits into the category of semisimple Lie algebras, i.e. Lie algebras whose adjoint representation is faithful. Examples are sl(n,R) and su(n), as well as direct sums of them. Let thus g be a semi-simple Lie algebra with adjoint representation ad, i.e.
$\mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}}):X\mapsto (\mathrm {ad} _{X}\colon Y\mapsto [X,Y])$.
Define now the Killing form
$k\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} :X\otimes Y\mapsto -\mathrm {tr} (\mathrm {ad} _{X}\circ \mathrm {ad} _{Y})$.
Due to the Cartan criterion, the Killing form is non-degenerate if and only if the Lie algebra is semisimple.
If g is in addition a simple Lie algebra, then the Killing form is up to rescaling the only invariant symmetric bilinear form.
References
This article incorporates material from Quadratic Lie algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Quadratic algebra
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
Definition
A graded quadratic algebra A is determined by a vector space of generators V = A1 and a subspace of homogeneous quadratic relations S ⊂ V ⊗ V (Polishchuk & Positselski 2005, p. 6). Thus
$A=T(V)/\langle S\rangle $
and inherits its grading from the tensor algebra T(V).
If the subspace of relations is instead allowed to also contain inhomogeneous degree 2 elements, i.e. S ⊂ k ⊕ V ⊕ (V ⊗ V), this construction results in a filtered quadratic algebra.
A graded quadratic algebra A as above admits a quadratic dual: the quadratic algebra generated by V* and with quadratic relations forming the orthogonal complement of S in V* ⊗ V*.
Examples
• Tensor algebra, symmetric algebra and exterior algebra of a finite-dimensional vector space are graded quadratic (in fact, Koszul) algebras.
• Universal enveloping algebra of a finite-dimensional Lie algebra is a filtered quadratic algebra.
References
• Polishchuk, Alexander; Positselski, Leonid (2005), Quadratic algebras, University Lecture Series, vol. 37, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3834-1, MR 2177131
• Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina (2009), "Quadratic duals, Koszul dual functors, and applications", Trans. Amer. Math. Soc., 361 (3): 1129–1172, arXiv:math.RT/0603475, doi:10.1090/S0002-9947-08-04539-X
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Quadratic assignment problem
The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and Beckmann.[1]
The problem models the following real-life problem:
There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows.
Intuitively, the cost function encourages facilities with high flows between each other to be placed close together.
The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequalities, hence the name.
Formal mathematical definition
The formal definition of the quadratic assignment problem is as follows:
Given two sets, P ("facilities") and L ("locations"), of equal size, together with a weight function w : P × P → R and a distance function d : L × L → R. Find the bijection f : P → L ("assignment") such that the cost function:
$\sum _{a,b\in P}w(a,b)\cdot d(f(a),f(b))$
is minimized.
Usually weight and distance functions are viewed as square real-valued matrices, so that the cost function is written down as:
$\sum _{a,b\in P}w_{a,b}d_{f(a),f(b)}$
In matrix notation:
$\min _{X\in \Pi _{n}}\operatorname {trace} (WXDX^{T})$
where $\Pi _{n}$ is the set of $n\times n$ permutation matrices, $W$ is the weight matrix and $D$ is the distance matrix.
Computational complexity
The problem is NP-hard, so there is no known algorithm for solving this problem in polynomial time, and even small instances may require long computation time. It was also proven that the problem does not have an approximation algorithm running in polynomial time for any (constant) factor, unless P = NP.[2] The travelling salesman problem (TSP) may be seen as a special case of QAP if one assumes that the flows connect all facilities only along a single ring, all flows have the same non-zero (constant) value and all distances are equal to the respective distances of the TSP instance. Many other problems of standard combinatorial optimization problems may be written in this form.
Applications
In addition to the original plant location formulation, QAP is a mathematical model for the problem of placement of interconnected electronic components onto a printed circuit board or on a microchip, which is part of the place and route stage of computer aided design in the electronics industry.
See also
• Quadratic bottleneck assignment problem
References
Notes
1. Koopmans TC, Beckmann M (1957). Assignment problems and the location of economic activities. Econometrica 25(1):53-76
2. Sahni, Sartaj; Gonzalez, Teofilo (July 1976). "P-Complete Approximation Problems". Journal of the ACM. 23 (3): 555–565. doi:10.1145/321958.321975. hdl:10338.dmlcz/103883.
Sources
• Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A2.5: ND43, pg.218.
External links
• https://www.opt.math.tugraz.at/qaplib/ QAPLIB - A Quadratic Assignment Problem Library
• http://www.wiomax.com/team/xie/maos-qap-quadratic-assignment-problem-project-portal/ MAOS-QAP - Java-based Quadratic Assignment Problem Solver
• https://CRAN.R-project.org/package=qap - R package qap: Heuristics for the Quadratic Assignment Problem
• https://apps.microsoft.com/store/detail/qapsolver/9N7WMCFB6NZZ - Metaheuristic QAP solver for Windows 10/11
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Quadratic bottleneck assignment problem
In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.[1]
It is related to the quadratic assignment problem in the same way as the linear bottleneck assignment problem is related to the linear assignment problem, the "sum" is replaced with "max" in the objective function.
The problem models the following real-life problem:
There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the maximum of the distances multiplied by the corresponding flows.
Computational complexity
The problem is NP-hard, as it can be used to formulate the Hamiltonian cycle problem by using flows in the pattern of a cycle and distances that are short for graph edges and long for non-edges.[2]
Special cases
• Bottleneck traveling salesman problem
• Graph bandwidth problem
References
1. Assignment Problems, by Rainer Burkard, Mauro Dell'Amico, Silvano Martello, 2009
2. Burkard, R. E.; Fincke, U. (1982), "On random quadratic bottleneck assignment problems", Mathematical Programming, 23 (2): 227–232, doi:10.1007/BF01583791, MR 0657082.
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Quadratic residue
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:
$x^{2}\equiv q{\pmod {n}}.$
Otherwise, q is called a quadratic nonresidue modulo n.
Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
History, conventions, and elementary facts
Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries established theorems[1] and formed conjectures[2] about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that if the context makes it clear, the adjective "quadratic" may be dropped.
For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, ..., n − 1. Because a2 ≡ (n − a)2 (mod n), the list of squares modulo n is symmetric around n/2, and the list only needs to go that high. This can be seen in the table below.
Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd).[3]
The product of two residues is always a residue.
Prime modulus
Modulo 2, every integer is a quadratic residue.
Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true for composite moduli.)[4]
Following this convention, the multiplicative inverse of a residue is a residue, and the inverse of a nonresidue is a nonresidue.[5]
Following this convention, modulo an odd prime number there are an equal number of residues and nonresidues.[4]
Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.[5]
The first supplement[6] to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p. This implies the following:
If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue.
If p ≡ 3 (mod 4) the negative of a residue modulo p is a nonresidue and the negative of a nonresidue is a residue.
Prime power modulus
All odd squares are ≡ 1 (mod 8) and thus also ≡ 1 (mod 4). If a is an odd number and m = 8, 16, or some higher power of 2, then a is a residue modulo m if and only if a ≡ 1 (mod 8).[7]
For example, mod (32) the odd squares are
12 ≡ 152 ≡ 1
32 ≡ 132 ≡ 9
52 ≡ 112 ≡ 25
72 ≡ 92 ≡ 49 ≡ 17
and the even ones are
02 ≡ 82 ≡ 162 ≡ 0
22 ≡ 62≡ 102 ≡ 142≡ 4
42 ≡ 122 ≡ 16.
So a nonzero number is a residue mod 8, 16, etc., if and only if it is of the form 4k(8n + 1).
A number a relatively prime to an odd prime p is a residue modulo any power of p if and only if it is a residue modulo p.[8]
If the modulus is pn,
then pka
is a residue modulo pn if k ≥ n
is a nonresidue modulo pn if k < n is odd
is a residue modulo pn if k < n is even and a is a residue
is a nonresidue modulo pn if k < n is even and a is a nonresidue.[9]
Notice that the rules are different for powers of two and powers of odd primes.
Modulo an odd prime power n = pk, the products of residues and nonresidues relatively prime to p obey the same rules as they do mod p; p is a nonresidue, and in general all the residues and nonresidues obey the same rules, except that the products will be zero if the power of p in the product ≥ n.
Modulo 8, the product of the nonresidues 3 and 5 is the nonresidue 7, and likewise for permutations of 3, 5 and 7. In fact, the multiplicative group of the non-residues and 1 form the Klein four-group.
Composite modulus not a prime power
The basic fact in this case is
if a is a residue modulo n, then a is a residue modulo pk for every prime power dividing n.
if a is a nonresidue modulo n, then a is a nonresidue modulo pk for at least one prime power dividing n.
Modulo a composite number, the product of two residues is a residue. The product of a residue and a nonresidue may be a residue, a nonresidue, or zero.
For example, from the table for modulus 6 1, 2, 3, 4, 5 (residues in bold).
The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2.
Also, the product of two nonresidues may be either a residue, a nonresidue, or zero.
For example, from the table for modulus 15 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (residues in bold).
The product of the nonresidues 2 and 8 is the residue 1, whereas the product of the nonresidues 2 and 7 is the nonresidue 14.
This phenomenon can best be described using the vocabulary of abstract algebra. The congruence classes relatively prime to the modulus are a group under multiplication, called the group of units of the ring Z/nZ, and the squares are a subgroup of it. Different nonresidues may belong to different cosets, and there is no simple rule that predicts which one their product will be in. Modulo a prime, there is only the subgroup of squares and a single coset.
The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring Z/nZ, which has zero divisors for composite n.
For this reason some authors[10] add to the definition that a quadratic residue a must not only be a square but must also be relatively prime to the modulus n. (a is coprime to n if and only if a2 is coprime to n.)
Although it makes things tidier, this article does not insist that residues must be coprime to the modulus.
Notations
Gauss[11] used R and N to denote residuosity and non-residuosity, respectively;
for example, 2 R 7 and 5 N 7, or 1 R 8 and 3 N 8.
Although this notation is compact and convenient for some purposes,[12][13] a more useful notation is the Legendre symbol, also called the quadratic character, which is defined for all integers a and positive odd prime numbers p as
$\left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{ if }}p{\text{ divides }}a\\+1&{\text{ if }}a\operatorname {R} p{\text{ and }}p{\text{ does not divide }}a\\-1&{\text{ if }}a\operatorname {N} p{\text{ and }}p{\text{ does not divide }}a\end{cases}}$
There are two reasons why numbers ≡ 0 (mod p) are treated specially. As we have seen, it makes many formulas and theorems easier to state. The other (related) reason is that the quadratic character is a homomorphism from the multiplicative group of nonzero congruence classes modulo p to the complex numbers under multiplication. Setting $({\tfrac {np}{p}})=0$ allows its domain to be extended to the multiplicative semigroup of all the integers.[14]
One advantage of this notation over Gauss's is that the Legendre symbol is a function that can be used in formulas.[15] It can also easily be generalized to cubic, quartic and higher power residues.[16]
There is a generalization of the Legendre symbol for composite values of p, the Jacobi symbol, but its properties are not as simple: if m is composite and the Jacobi symbol $({\tfrac {a}{m}})=-1,$ then a N m, and if a R m then $({\tfrac {a}{m}})=1,$ but if $({\tfrac {a}{m}})=1$ we do not know whether a R m or a N m. For example: $({\tfrac {2}{15}})=1$ and $({\tfrac {4}{15}})=1$, but 2 N 15 and 4 R 15. If m is prime, the Jacobi and Legendre symbols agree.
Distribution of quadratic residues
Although quadratic residues appear to occur in a rather random pattern modulo n, and this has been exploited in such applications as acoustics and cryptography, their distribution also exhibits some striking regularities.
Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem (CRT) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, ..., M are all residues modulo p.
For example, if p ≡ 1 (mod 8), (mod 12), (mod 5) and (mod 28), then by the law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo p, and thus all numbers 1–10 will be. The CRT says that this is the same as p ≡ 1 (mod 840), and Dirichlet's theorem says there are an infinite number of primes of this form. 2521 is the smallest, and indeed 12 ≡ 1, 10462 ≡ 2, 1232 ≡ 3, 22 ≡ 4, 6432 ≡ 5, 872 ≡ 6, 6682 ≡ 7, 4292 ≡ 8, 32 ≡ 9, and 5292 ≡ 10 (mod 2521).
Dirichlet's formulas
The first of these regularities stems from Peter Gustav Lejeune Dirichlet's work (in the 1830s) on the analytic formula for the class number of binary quadratic forms.[17] Let q be a prime number, s a complex variable, and define a Dirichlet L-function as
$L(s)=\sum _{n=1}^{\infty }\left({\frac {n}{q}}\right)n^{-s}.$
Dirichlet showed that if q ≡ 3 (mod 4), then
$L(1)=-{\frac {\pi }{\sqrt {q}}}\sum _{n=1}^{q-1}{\frac {n}{q}}\left({\frac {n}{q}}\right)>0.$
Therefore, in this case (prime q ≡ 3 (mod 4)), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, ..., q − 1 is a negative number.
For example, modulo 11,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (residues in bold)
1 + 4 + 9 + 5 + 3 = 22, 2 + 6 + 7 + 8 + 10 = 33, and the difference is −11.
In fact the difference will always be an odd multiple of q if q > 3.[18] In contrast, for prime q ≡ 1 (mod 4), the sum of the quadratic residues minus the sum of the nonresidues in the range 1, 2, ..., q − 1 is zero, implying that both sums equal ${\frac {q(q-1)}{4}}$.
Dirichlet also proved that for prime q ≡ 3 (mod 4),
$L(1)={\frac {\pi }{\left(2-\left({\frac {2}{q}}\right)\right)\!{\sqrt {q}}}}\sum _{n=1}^{\frac {q-1}{2}}\left({\frac {n}{q}}\right)>0.$
This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (q − 1)/2.
For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, and 5), but only one nonresidue (2).
An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.[19]
Law of quadratic reciprocity
Main article: quadratic reciprocity
If p and q are odd primes, then:
((p is a quadratic residue mod q) if and only if (q is a quadratic residue mod p)) if and only if (at least one of p and q is congruent to 1 mod 4).
That is:
$\left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}\cdot {\frac {q-1}{2}}}$
where $\left({\frac {p}{q}}\right)$ is the Legendre symbol.
Thus, for numbers a and odd primes p that don't divide a:
a a is a quadratic residue mod p if and only if a a is a quadratic residue mod p if and only if
1 (every prime p) −1 p ≡ 1 (mod 4)
2 p ≡ 1, 7 (mod 8) −2 p ≡ 1, 3 (mod 8)
3 p ≡ 1, 11 (mod 12) −3 p ≡ 1 (mod 3)
4 (every prime p) −4 p ≡ 1 (mod 4)
5 p ≡ 1, 4 (mod 5) −5 p ≡ 1, 3, 7, 9 (mod 20)
6 p ≡ 1, 5, 19, 23 (mod 24) −6 p ≡ 1, 5, 7, 11 (mod 24)
7 p ≡ 1, 3, 9, 19, 25, 27 (mod 28) −7 p ≡ 1, 2, 4 (mod 7)
8 p ≡ 1, 7 (mod 8) −8 p ≡ 1, 3 (mod 8)
9 (every prime p) −9 p ≡ 1 (mod 4)
10 p ≡ 1, 3, 9, 13, 27, 31, 37, 39 (mod 40) −10 p ≡ 1, 7, 9, 11, 13, 19, 23, 37 (mod 40)
11 p ≡ 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44) −11 p ≡ 1, 3, 4, 5, 9 (mod 11)
12 p ≡ 1, 11 (mod 12) −12 p ≡ 1 (mod 3)
Pairs of residues and nonresidues
Modulo a prime p, the number of pairs n, n + 1 where n R p and n + 1 R p, or n N p and n + 1 R p, etc., are almost equal. More precisely,[20][21] let p be an odd prime. For i, j = 0, 1 define the sets
$A_{ij}=\left\{k\in \{1,2,\dots ,p-2\}:\left({\frac {k}{p}}\right)=(-1)^{i}\land \left({\frac {k+1}{p}}\right)=(-1)^{j}\right\},$
and let
$\alpha _{ij}=|A_{ij}|.$
That is,
α00 is the number of residues that are followed by a residue,
α01 is the number of residues that are followed by a nonresidue,
α10 is the number of nonresidues that are followed by a residue, and
α11 is the number of nonresidues that are followed by a nonresidue.
Then if p ≡ 1 (mod 4)
$\alpha _{00}={\frac {p-5}{4}},\;\alpha _{01}=\alpha _{10}=\alpha _{11}={\frac {p-1}{4}}$
and if p ≡ 3 (mod 4)
$\alpha _{01}={\frac {p+1}{4}},\;\alpha _{00}=\alpha _{10}=\alpha _{11}={\frac {p-3}{4}}.$
For example: (residues in bold)
Modulo 17
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
A00 = {1,8,15},
A01 = {2,4,9,13},
A10 = {3,7,12,14},
A11 = {5,6,10,11}.
Modulo 19
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
A00 = {4,5,6,16},
A01 = {1,7,9,11,17},
A10 = {3,8,10,15},
A11 = {2,12,13,14}.
Gauss (1828)[22] introduced this sort of counting when he proved that if p ≡ 1 (mod 4) then x4 ≡ 2 (mod p) can be solved if and only if p = a2 + 64 b2.
The Pólya–Vinogradov inequality
The values of $({\tfrac {a}{p}})$ for consecutive values of a mimic a random variable like a coin flip.[23] Specifically, Pólya and Vinogradov proved[24] (independently) in 1918 that for any nonprincipal Dirichlet character χ(n) modulo q and any integers M and N,
$\left|\sum _{n=M+1}^{M+N}\chi (n)\right|=O\left({\sqrt {q}}\log q\right),$
in big O notation. Setting
$\chi (n)=\left({\frac {n}{q}}\right),$
this shows that the number of quadratic residues modulo q in any interval of length N is
${\frac {1}{2}}N+O({\sqrt {q}}\log q).$
It is easy[25] to prove that
$\left|\sum _{n=M+1}^{M+N}\left({\frac {n}{q}}\right)\right|<{\sqrt {q}}\log q.$
In fact,[26]
$\left|\sum _{n=M+1}^{M+N}\left({\frac {n}{q}}\right)\right|<{\frac {4}{\pi ^{2}}}{\sqrt {q}}\log q+0.41{\sqrt {q}}+0.61.$
Montgomery and Vaughan improved this in 1977, showing that, if the generalized Riemann hypothesis is true then
$\left|\sum _{n=M+1}^{M+N}\chi (n)\right|=O\left({\sqrt {q}}\log \log q\right).$
This result cannot be substantially improved, for Schur had proved in 1918 that
$\max _{N}\left|\sum _{n=1}^{N}\left({\frac {n}{q}}\right)\right|>{\frac {1}{2\pi }}{\sqrt {q}}$
and Paley had proved in 1932 that
$\max _{N}\left|\sum _{n=1}^{N}\left({\frac {d}{n}}\right)\right|>{\frac {1}{7}}{\sqrt {d}}\log \log d$
for infinitely many d > 0.
Least quadratic non-residue
The least quadratic residue mod p is clearly 1. The question of the magnitude of the least quadratic non-residue n(p) is more subtle, but it is always prime, with 7 appearing for the first time at 71.
The Pólya–Vinogradov inequality above gives O(√p log p).
The best unconditional estimate is n(p) ≪ pθ for any θ>1/4√e, obtained by estimates of Burgess on character sums.[27]
Assuming the Generalised Riemann hypothesis, Ankeny obtained n(p) ≪ (log p)2.[28]
Linnik showed that the number of p less than X such that n(p) > Xε is bounded by a constant depending on ε.[27]
The least quadratic non-residues mod p for odd primes p are:
2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, ... (sequence A053760 in the OEIS)
Quadratic excess
Let p be an odd prime. The quadratic excess E(p) is the number of quadratic residues on the range (0,p/2) minus the number in the range (p/2,p) (sequence A178153 in the OEIS). For p congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under r ↔ p−r. For p congruent to 3 mod 4, the excess E is always positive.[29]
Complexity of finding square roots
That is, given a number a and a modulus n, how hard is it
1. to tell whether an x solving x2 ≡ a (mod n) exists
2. assuming one does exist, to calculate it?
An important difference between prime and composite moduli shows up here. Modulo a prime p, a quadratic residue a has 1 + (a|p) roots (i.e. zero if a N p, one if a ≡ 0 (mod p), or two if a R p and gcd(a,p) = 1.)
In general if a composite modulus n is written as a product of powers of distinct primes, and there are n1 roots modulo the first one, n2 mod the second, ..., there will be n1n2... roots modulo n.
The theoretical way solutions modulo the prime powers are combined to make solutions modulo n is called the Chinese remainder theorem; it can be implemented with an efficient algorithm.[30]
For example:
Solve x2 ≡ 6 (mod 15).
x2 ≡ 6 (mod 3) has one solution, 0; x2 ≡ 6 (mod 5) has two, 1 and 4.
and there are two solutions modulo 15, namely 6 and 9.
Solve x2 ≡ 4 (mod 15).
x2 ≡ 4 (mod 3) has two solutions, 1 and 2; x2 ≡ 4 (mod 5) has two, 2 and 3.
and there are four solutions modulo 15, namely 2, 7, 8, and 13.
Solve x2 ≡ 7 (mod 15).
x2 ≡ 7 (mod 3) has two solutions, 1 and 2; x2 ≡ 7 (mod 5) has no solutions.
and there are no solutions modulo 15.
Prime or prime power modulus
First off, if the modulus n is prime the Legendre symbol $\left({\frac {a}{n}}\right)$ can be quickly computed using a variation of Euclid's algorithm[31] or the Euler's criterion. If it is −1 there is no solution. Secondly, assuming that $\left({\frac {a}{n}}\right)=1$, if n ≡ 3 (mod 4), Lagrange found that the solutions are given by
$x\equiv \pm \;a^{(n+1)/4}{\pmod {n}},$
and Legendre found a similar solution[32] if n ≡ 5 (mod 8):
$x\equiv {\begin{cases}\pm \;a^{(n+3)/8}{\pmod {n}}&{\text{ if }}a{\text{ is a quartic residue modulo }}n\\\pm \;a^{(n+3)/8}2^{(n-1)/4}{\pmod {n}}&{\text{ if }}a{\text{ is a quartic non-residue modulo }}n\end{cases}}$
For prime n ≡ 1 (mod 8), however, there is no known formula. Tonelli[33] (in 1891) and Cipolla[34] found efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue modulo n, and there is no efficient deterministic algorithm known for doing that. But since half the numbers between 1 and n are nonresidues, picking numbers x at random and calculating the Legendre symbol $\left({\frac {x}{n}}\right)$ until a nonresidue is found will quickly produce one. A slight variant of this algorithm is the Tonelli–Shanks algorithm.
If the modulus n is a prime power n = pe, a solution may be found modulo p and "lifted" to a solution modulo n using Hensel's lemma or an algorithm of Gauss.[8]
Composite modulus
If the modulus n has been factored into prime powers the solution was discussed above.
If n is not congruent to 2 modulo 4 and the Kronecker symbol $\left({\tfrac {a}{n}}\right)=-1$ then there is no solution; if n is congruent to 2 modulo 4 and $\left({\tfrac {a}{n/2}}\right)=-1$, then there is also no solution. If n is not congruent to 2 modulo 4 and $\left({\tfrac {a}{n}}\right)=1$, or n is congruent to 2 modulo 4 and $\left({\tfrac {a}{n/2}}\right)=1$, there may or may not be one.
If the complete factorization of n is not known, and $\left({\tfrac {a}{n}}\right)=1$ and n is not congruent to 2 modulo 4, or n is congruent to 2 modulo 4 and $\left({\tfrac {a}{n/2}}\right)=1$, the problem is known to be equivalent to integer factorization of n (i.e. an efficient solution to either problem could be used to solve the other efficiently).
The above discussion indicates how knowing the factors of n allows us to find the roots efficiently. Say there were an efficient algorithm for finding square roots modulo a composite number. The article congruence of squares discusses how finding two numbers x and y where x2 ≡ y2 (mod n) and x ≠ ±y suffices to factorize n efficiently. Generate a random number, square it modulo n, and have the efficient square root algorithm find a root. Repeat until it returns a number not equal to the one we originally squared (or its negative modulo n), then follow the algorithm described in congruence of squares. The efficiency of the factoring algorithm depends on the exact characteristics of the root-finder (e.g. does it return all roots? just the smallest one? a random one?), but it will be efficient.[35]
Determining whether a is a quadratic residue or nonresidue modulo n (denoted a R n or a N n) can be done efficiently for prime n by computing the Legendre symbol. However, for composite n, this forms the quadratic residuosity problem, which is not known to be as hard as factorization, but is assumed to be quite hard.
On the other hand, if we want to know if there is a solution for x less than some given limit c, this problem is NP-complete;[36] however, this is a fixed-parameter tractable problem, where c is the parameter.
In general, to determine if a is a quadratic residue modulo composite n, one can use the following theorem:[37]
Let n > 1, and gcd(a,n) = 1. Then x2 ≡ a (mod n) is solvable if and only if:
• The Legendre symbol $\left({\tfrac {a}{p}}\right)=1$ for all odd prime divisors p of n.
• a ≡ 1 (mod 4) if n is divisible by 4 but not 8; or a ≡ 1 (mod 8) if n is divisible by 8.
Note: This theorem essentially requires that the factorization of n is known. Also notice that if gcd(a,n) = m, then the congruence can be reduced to a/m ≡ x2/m (mod n/m), but then this takes the problem away from quadratic residues (unless m is a square).
The number of quadratic residues
The list of the number of quadratic residues modulo n, for n = 1, 2, 3 ..., looks like:
1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, ... (sequence A000224 in the OEIS)
A formula to count the number of squares modulo n is given by Stangl.[38]
Applications of quadratic residues
Acoustics
Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.[39]
Graph theory
Paley graphs are dense undirected graphs, one for each prime p ≡ 1 (mod 4), that form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices.
Paley digraphs are directed analogs of Paley graphs, one for each p ≡ 3 (mod 4), that yield antisymmetric conference matrices.
The construction of these graphs uses quadratic residues.
Cryptography
The fact that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed to be a hard problem) has been used for constructing cryptographic schemes such as the Rabin cryptosystem and the oblivious transfer. The quadratic residuosity problem is the basis for the Goldwasser-Micali cryptosystem.
The discrete logarithm is a similar problem that is also used in cryptography.
Primality testing
Euler's criterion is a formula for the Legendre symbol (a|p) where p is prime. If p is composite the formula may or may not compute (a|p) correctly. The Solovay–Strassen primality test for whether a given number n is prime or composite picks a random a and computes (a|n) using a modification of Euclid's algorithm,[40] and also using Euler's criterion.[41] If the results disagree, n is composite; if they agree, n may be composite or prime. For a composite n at least 1/2 the values of a in the range 2, 3, ..., n − 1 will return "n is composite"; for prime n none will. If, after using many different values of a, n has not been proved composite it is called a "probable prime".
The Miller–Rabin primality test is based on the same principles. There is a deterministic version of it, but the proof that it works depends on the generalized Riemann hypothesis; the output from this test is "n is definitely composite" or "either n is prime or the GRH is false". If the second output ever occurs for a composite n, then the GRH would be false, which would have implications through many branches of mathematics.
Integer factorization
In § VI of the Disquisitiones Arithmeticae[42] Gauss discusses two factoring algorithms that use quadratic residues and the law of quadratic reciprocity.
Several modern factorization algorithms (including Dixon's algorithm, the continued fraction method, the quadratic sieve, and the number field sieve) generate small quadratic residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.
Table of quadratic residues
The following table (sequence A096008 in the OEIS) lists the quadratic residues mod 1 to 75 (a red number means it is not coprime to n). (For the quadratic residues coprime to n, see OEIS: A096103, and for nonzero quadratic residues, see OEIS: A046071.)
n quadratic residues mod n n quadratic residues mod n n quadratic residues mod n
1 0 26 0, 1, 3, 4, 9, 10, 12, 13, 14, 16, 17, 22, 23, 25 51 0, 1, 4, 9, 13, 15, 16, 18, 19, 21, 25, 30, 33, 34, 36, 42, 43, 49
2 0, 1 27 0, 1, 4, 7, 9, 10, 13, 16, 19, 22, 25 52 0, 1, 4, 9, 12, 13, 16, 17, 25, 29, 36, 40, 48, 49
3 0, 1 28 0, 1, 4, 8, 9, 16, 21, 25 53 0, 1, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 24, 25, 28, 29, 36, 37, 38, 40, 42, 43, 44, 46, 47, 49, 52
4 0, 1 29 0, 1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28 54 0, 1, 4, 7, 9, 10, 13, 16, 19, 22, 25, 27, 28, 31, 34, 36, 37, 40, 43, 46, 49, 52
5 0, 1, 4 30 0, 1, 4, 6, 9, 10, 15, 16, 19, 21, 24, 25 55 0, 1, 4, 5, 9, 11, 14, 15, 16, 20, 25, 26, 31, 34, 36, 44, 45, 49
6 0, 1, 3, 4 31 0, 1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28 56 0, 1, 4, 8, 9, 16, 25, 28, 32, 36, 44, 49
7 0, 1, 2, 4 32 0, 1, 4, 9, 16, 17, 25 57 0, 1, 4, 6, 7, 9, 16, 19, 24, 25, 28, 30, 36, 39, 42, 43, 45, 49, 54, 55
8 0, 1, 4 33 0, 1, 3, 4, 9, 12, 15, 16, 22, 25, 27, 31 58 0, 1, 4, 5, 6, 7, 9, 13, 16, 20, 22, 23, 24, 25, 28, 29, 30, 33, 34, 35, 36, 38, 42, 45, 49, 51, 52, 53, 54, 57
9 0, 1, 4, 7 34 0, 1, 2, 4, 8, 9, 13, 15, 16, 17, 18, 19, 21, 25, 26, 30, 32, 33 59 0, 1, 3, 4, 5, 7, 9, 12, 15, 16, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 35, 36, 41, 45, 46, 48, 49, 51, 53, 57
10 0, 1, 4, 5, 6, 9 35 0, 1, 4, 9, 11, 14, 15, 16, 21, 25, 29, 30 60 0, 1, 4, 9, 16, 21, 24, 25, 36, 40, 45, 49
11 0, 1, 3, 4, 5, 9 36 0, 1, 4, 9, 13, 16, 25, 28 61 0, 1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 52, 56, 57, 58, 60
12 0, 1, 4, 9 37 0, 1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36 62 0, 1, 2, 4, 5, 7, 8, 9, 10, 14, 16, 18, 19, 20, 25, 28, 31, 32, 33, 35, 36, 38, 39, 40, 41, 45, 47, 49, 50, 51, 56, 59
13 0, 1, 3, 4, 9, 10, 12 38 0, 1, 4, 5, 6, 7, 9, 11, 16, 17, 19, 20, 23, 24, 25, 26, 28, 30, 35, 36 63 0, 1, 4, 7, 9, 16, 18, 22, 25, 28, 36, 37, 43, 46, 49, 58
14 0, 1, 2, 4, 7, 8, 9, 11 39 0, 1, 3, 4, 9, 10, 12, 13, 16, 22, 25, 27, 30, 36 64 0, 1, 4, 9, 16, 17, 25, 33, 36, 41, 49, 57
15 0, 1, 4, 6, 9, 10 40 0, 1, 4, 9, 16, 20, 24, 25, 36 65 0, 1, 4, 9, 10, 14, 16, 25, 26, 29, 30, 35, 36, 39, 40, 49, 51, 55, 56, 61, 64
16 0, 1, 4, 9 41 0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40 66 0, 1, 3, 4, 9, 12, 15, 16, 22, 25, 27, 31, 33, 34, 36, 37, 42, 45, 48, 49, 55, 58, 60, 64
17 0, 1, 2, 4, 8, 9, 13, 15, 16 42 0, 1, 4, 7, 9, 15, 16, 18, 21, 22, 25, 28, 30, 36, 37, 39 67 0, 1, 4, 6, 9, 10, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 29, 33, 35, 36, 37, 39, 40, 47, 49, 54, 55, 56, 59, 60, 62, 64, 65
18 0, 1, 4, 7, 9, 10, 13, 16 43 0, 1, 4, 6, 9, 10, 11, 13, 14, 15, 16, 17, 21, 23, 24, 25, 31, 35, 36, 38, 40, 41 68 0, 1, 4, 8, 9, 13, 16, 17, 21, 25, 32, 33, 36, 49, 52, 53, 60, 64
19 0, 1, 4, 5, 6, 7, 9, 11, 16, 17 44 0, 1, 4, 5, 9, 12, 16, 20, 25, 33, 36, 37 69 0, 1, 3, 4, 6, 9, 12, 13, 16, 18, 24, 25, 27, 31, 36, 39, 46, 48, 49, 52, 54, 55, 58, 64
20 0, 1, 4, 5, 9, 16 45 0, 1, 4, 9, 10, 16, 19, 25, 31, 34, 36, 40 70 0, 1, 4, 9, 11, 14, 15, 16, 21, 25, 29, 30, 35, 36, 39, 44, 46, 49, 50, 51, 56, 60, 64, 65
21 0, 1, 4, 7, 9, 15, 16, 18 46 0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 23, 24, 25, 26, 27, 29, 31, 32, 35, 36, 39, 41 71 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 25, 27, 29, 30, 32, 36, 37, 38, 40, 43, 45, 48, 49, 50, 54, 57, 58, 60, 64
22 0, 1, 3, 4, 5, 9, 11, 12, 14, 15, 16, 20 47 0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 17, 18, 21, 24, 25, 27, 28, 32, 34, 36, 37, 42 72 0, 1, 4, 9, 16, 25, 28, 36, 40, 49, 52, 64
23 0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18 48 0, 1, 4, 9, 16, 25, 33, 36 73 0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 19, 23, 24, 25, 27, 32, 35, 36, 37, 38, 41, 46, 48, 49, 50, 54, 55, 57, 61, 64, 65, 67, 69, 70, 71, 72
24 0, 1, 4, 9, 12, 16 49 0, 1, 2, 4, 8, 9, 11, 15, 16, 18, 22, 23, 25, 29, 30, 32, 36, 37, 39, 43, 44, 46 74 0, 1, 3, 4, 7, 9, 10, 11, 12, 16, 21, 25, 26, 27, 28, 30, 33, 34, 36, 37, 38, 40, 41, 44, 46, 47, 48, 49, 53, 58, 62, 63, 64, 65, 67, 70, 71, 73
25 0, 1, 4, 6, 9, 11, 14, 16, 19, 21, 24 50 0, 1, 4, 6, 9, 11, 14, 16, 19, 21, 24, 25, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49 75 0, 1, 4, 6, 9, 16, 19, 21, 24, 25, 31, 34, 36, 39, 46, 49, 51, 54, 61, 64, 66, 69
Quadratic Residues (see also A048152, A343720)
x12345678910111213141516171819202122232425
x2 1 4 9 16 25 36 49 64 81 100121144169196225256289324361400441484529576625
mod 1 00000 00000 00000 00000 00000
mod 2 10 10 10 10 10 10 10 10 10 10 10 10 1
mod 3 110 110 110 110 110 110 110 110 1
mod 4 10 10 10 10 10 10 10 10 10 10 10 10 1
mod 5 14410 14410 14410 14410 14410
mod 6 143410 143410 143410 143410 1
mod 7 1422410 1422410 1422410 1422
mod 8 1410 1410 1410 1410 1410 1410 1
mod 9 140770410 140770410 1407704
mod 10 1496569410 1496569410 14965
mod 11 14953359410 14953359410 149
mod 12 149410 149410 149410 149410 1
mod 13 14931210101239410 1493121010123941
mod 14 1492118781129410 1492118781129
mod 15 14911064461019410 149110644610
mod 16 14909410 14909410 14909410 1
mod 17 14916821513131528169410 14916821513
mod 18 149167013109101307169410 149167013
mod 19 1491661711755711176169410 14916617
mod 20 149165169410 149165169410 149165
mod 21 14916415711816161817154169410 14916
mod 22 149163145201512111215205143169410 149
mod 23 1491621331812866812183132169410 14
mod 24 149161121169410 149161121169410 1
mod 25 1491601124146021191921061424110169410
See also
• Euler's criterion
• Gauss's lemma
• Zolotarev's lemma
• Character sum
• Law of quadratic reciprocity
• Quadratic residue code
Notes
1. Lemmemeyer, Ch. 1
2. Lemmermeyer, pp 6–8, p. 16 ff
3. Gauss, DA, art. 94
4. Gauss, DA, art. 96
5. Gauss, DA, art. 98
6. Gauss, DA, art 111
7. Gauss, DA, art. 103
8. Gauss, DA, art. 101
9. Gauss, DA, art. 102
10. e.g., Ireland & Rosen 1990, p. 50
11. Gauss, DA, art. 131
12. e.g. Hardy and Wright use it
13. Gauss, DA, art 230 ff.
14. This extension of the domain is necessary for defining L functions.
15. See Legendre symbol#Properties of the Legendre symbol for examples
16. Lemmermeyer, pp 111–end
17. Davenport 2000, pp. 8–9, 43–51. These are classical results.
18. Davenport 2000, pp. 49–51, (conjectured by Jacobi, proved by Dirichlet)
19. Davenport 2000, p. 9
20. Lemmermeyer, p. 29 ex. 1.22; cf pp. 26–27, Ch. 10
21. Crandall & Pomerance, ex 2.38, pp 106–108
22. Gauss, Theorie der biquadratischen Reste, Erste Abhandlung (pp 511–533 of the Untersuchungen über hohere Arithmetik)
23. Crandall & Pomerance, ex 2.38, pp 106–108 discuss the similarities and differences. For example, tossing n coins, it is possible (though unlikely) to get n/2 heads followed by that many tails. V-P inequality rules that out for residues.
24. Davenport 2000, pp. 135–137, (proof of P–V, (in fact big-O can be replaced by 2); journal references for Paley, Montgomery, and Schur)
25. Planet Math: Proof of Pólya–Vinogradov Inequality in external links. The proof is a page long and only requires elementary facts about Gaussian sums
26. Pomerance & Crandall, ex 2.38 pp.106–108. result from T. Cochrane, "On a trigonometric inequality of Vinogradov", J. Number Theory, 27:9–16, 1987
27. Friedlander, John B.; Iwaniec, Henryk (2010). Opera De Cribro. American Mathematical Society. p. 156. ISBN 978-0-8218-4970-5. Zbl 1226.11099.
28. Montgomery, Hugh L. (1994). Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis. American Mathematical Society. p. 176. ISBN 0-8218-0737-4. Zbl 0814.11001.
29. Bateman, Paul T.; Diamond, Harold G. (2004). Analytic Number Theory. World Scientific. p. 250. ISBN 981-256-080-7. Zbl 1074.11001.
30. Bach & Shallit 1996, p. 104 ff; it requires O(log2 m) steps where m is the number of primes dividing n.
31. Bach & Shallit 1996, p. 113; computing $\left({\frac {a}{n}}\right)$ requires O(log a log n) steps
32. Lemmermeyer, p. 29
33. Bach & Shallit 1996, p. 156 ff; the algorithm requires O(log4n) steps.
34. Bach & Shallit 1996, p. 156 ff; the algorithm requires O(log3 n) steps and is also nondeterministic.
35. Crandall & Pomerance, ex. 6.5 & 6.6, p.273
36. Manders & Adleman 1978
37. Burton, David (2007). Elementary Number Theory. New York: McGraw HIll. p. 195.
38. Stangl, Walter D. (October 1996), "Counting Squares in ℤn" (PDF), Mathematics Magazine, 69 (4): 285–289, doi:10.2307/2690536, JSTOR 2690536
39. Walker, R. "The design and application of modular acoustic diffusing elements" (PDF). BBC Research Department. Retrieved 25 October 2016.
40. Bach & Shallit 1996, p. 113
41. Bach & Shallit 1996, pp. 109–110; Euler's criterion requires O(log3 n) steps
42. Gauss, DA, arts 329–334
References
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
• Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second corrected ed.), New York: Springer, ISBN 0-387-96254-9 {{citation}}: |first2= has generic name (help)
• Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen über hohere Arithmetik [Disquisitiones Arithemeticae & other papers on number theory] (second ed.), New York: Chelsea, ISBN 0-8284-0191-8 {{citation}}: |first2= has generic name (help)
• Bach, Eric; Shallit, Jeffrey (1996), Efficient Algorithms, Algorithmic Number Theory, vol. I, Cambridge: The MIT Press, ISBN 0-262-02405-5
• Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 0-387-94777-9
• Davenport, Harold (2000), Multiplicative Number Theory (third ed.), New York: Springer, ISBN 0-387-95097-4
• Garey, Michael R.; Johnson, David S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, ISBN 0-7167-1045-5 A7.1: AN1, pg.249.
• Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
• Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (second ed.), New York: Springer, ISBN 0-387-97329-X
• Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66957-4
• Manders, Kenneth L.; Adleman, Leonard (1978), "NP-Complete Decision Problems for Binary Quadratics", Journal of Computer and System Sciences, 16 (2): 168–184, doi:10.1016/0022-0000(78)90044-2.
External links
• Weisstein, Eric W. "Quadratic Residue". MathWorld.
• Proof of Pólya–Vinogradov inequality at PlanetMath.
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Rate of convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence $(x_{n})$ that converges to $x^{*}$ is said to have order of convergence $q\geq 1$ and rate of convergence $\mu $ if
$\lim _{n\rightarrow \infty }{\frac {\left|x_{n+1}-x^{*}\right|}{\left|x_{n}-x^{*}\right|^{q}}}=\mu .$[1]
Differential equations
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The rate of convergence $\mu $ is also called the asymptotic error constant. Note that this terminology is not standardized and some authors will use rate where this article uses order (e.g., [2]).
In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence.
Similar concepts are used for discretization methods. The solution of the discretized problem converges to the solution of the continuous problem as the grid size goes to zero, and the speed of convergence is one of the factors of the efficiency of the method. However, the terminology, in this case, is different from the terminology for iterative methods.
Series acceleration is a collection of techniques for improving the rate of convergence of a series discretization. Such acceleration is commonly accomplished with sequence transformations.
Convergence speed for iterative methods
Convergence definitions
Suppose that the sequence $(x_{k})$ converges to the number $L$. The sequence is said to converge with order $q$ to $L$, and with a rate of convergence[3] of $\mu $, if
$\lim _{k\to \infty }{\frac {|x_{k+1}-L|}{|x_{k}-L|^{q}}}=\mu \qquad {\text{(Definition 1)}}$
for some positive constant $\mu \in (0,\infty )$ if $q>1$, and $\mu \in (0,1)$ if $q=1$.[4][5] It is not necessary, however, that $q$ be an integer. For example, the secant method, when converging to a regular, simple root, has an order of φ ≈ 1.618.
Convergence with order
• $q=1$ is called linear convergence if $\mu \in (0,1)$, and the sequence is said to converge Q-linearly to $L$.
• $q=2$ is called quadratic convergence.
• $q=3$ is called cubic convergence.
• etc.
Order estimation
A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to $q$:
$q\approx {\frac {\log \left|{\frac {x_{k+1}-x_{k}}{x_{k}-x_{k-1}}}\right|}{\log \left|{\frac {x_{k}-x_{k-1}}{x_{k-1}-x_{k-2}}}\right|}}.$[6]
Q-convergence definitions
In addition to the previously defined Q-linear convergence, a few other Q-convergence definitions exist. Given Definition 1 defined above, the sequence is said to converge Q-superlinearly to $L$ (i.e. faster than linearly) in all the cases where $q>1$ and also the case $q=1,\mu =0$.[7] Given Definition 1, the sequence is said to converge Q-sublinearly to $L$ (i.e. slower than linearly) if $q=1,\mu =1$. The sequence $(x_{k})$ converges logarithmically to $L$ if the sequence converges sublinearly and additionally if
$\lim _{k\to \infty }{\frac {|x_{k+2}-x_{k+1}|}{|x_{k+1}-x_{k}|}}=1$.[8]
Note that unlike previous definitions, logarithmic convergence is not called "Q-logarithmic."
In the definitions above, the "Q-" stands for "quotient" because the terms are defined using the quotient between two successive terms.[9]: 619 Often, however, the "Q-" is dropped and a sequence is simply said to have linear convergence, quadratic convergence, etc.
R-convergence definition
The Q-convergence definitions have a shortcoming in that they do not include some sequences, such as the sequence $(b_{k})$ below, which converge reasonably fast, but whose rate is variable. Therefore, the definition of rate of convergence is extended as follows.
Suppose that $(x_{k})$ converges to $L$. The sequence is said to converge R-linearly to $L$ if there exists a sequence $(\varepsilon _{k})$ such that
$|x_{k}-L|\leq \varepsilon _{k}\quad {\text{for all }}k\,,$
and $(\varepsilon _{k})$ converges Q-linearly to zero.[3] The "R-" prefix stands for "root". [9]: 620
Examples
Consider the sequence
$(a_{k})=\left\{1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},{\frac {1}{16}},{\frac {1}{32}},\ldots ,{\frac {1}{2^{k}}},...\right\}.$
It can be shown that this sequence converges to $L=0$. To determine the type of convergence, we plug the sequence into the definition of Q-linear convergence,
$\lim _{k\to \infty }{\frac {\left|1/2^{k+1}-0\right|}{\left|1/2^{k}-0\right|}}=\lim _{k\to \infty }{\frac {2^{k}}{2^{k+1}}}={\frac {1}{2}}.$
Thus, we find that $(a_{k})$ converges Q-linearly and has a convergence rate of $\mu =1/2$. More generally, for any $c\in \mathbb {R} ,\mu \in (-1,1)$, the sequence $(c\mu ^{k})$ converges linearly with rate $|\mu |$.
The sequence
$(b_{k})=\left\{1,1,{\frac {1}{4}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{16}},\ldots ,{\frac {1}{4^{\left\lfloor {\frac {k}{2}}\right\rfloor }}},\,\ldots \right\}$
also converges linearly to 0 with rate 1/2 under the R-convergence definition, but not under the Q-convergence definition. (Note that $\lfloor x\rfloor $ is the floor function, which gives the largest integer that is less than or equal to $x$.)
The sequence
$(c_{k})=\left\{{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{16}},{\frac {1}{256}},{\frac {1}{65,\!536}},\ldots ,{\frac {1}{2^{2^{k}}}},\ldots \right\}$
converges superlinearly. In fact, it is quadratically convergent.
Finally, the sequence
$(d_{k})=\left\{1,{\frac {1}{2}},{\frac {1}{3}},{\frac {1}{4}},{\frac {1}{5}},{\frac {1}{6}},\ldots ,{\frac {1}{k+1}},\ldots \right\}$
converges sublinearly and logarithmically.
Convergence speed for discretization methods
A similar situation exists for discretization methods designed to approximate a function $y=f(x)$, which might be an integral being approximated by numerical quadrature, or the solution of an ordinary differential equation (see example below). The discretization method generates a sequence ${y_{0},y_{1},y_{2},y_{3},...}$, where each successive $y_{j}$ is a function of $y_{j-1},y_{j-2},...$ along with the grid spacing $h$ between successive values of the independent variable $x$. The important parameter here for the convergence speed to $y=f(x)$ is the grid spacing $h$, inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of $x$.
In this case, the sequence $(y_{n})$ is said to converge to the sequence $f(x_{n})$ with order q if there exists a constant C such that
$|y_{n}-f(x_{n})|<Ch^{q}{\text{ for all }}n.$
This is written as $|y_{n}-f(x_{n})|={\mathcal {O}}(h^{q})$ using big O notation.
This is the relevant definition when discussing methods for numerical quadrature or the solution of ordinary differential equations (ODEs).
A practical method to estimate the order of convergence for a discretization method is pick step sizes $h_{\text{new}}$ and $h_{\text{old}}$ and calculate the resulting errors $e_{\text{new}}$ and $e_{\text{old}}$. The order of convergence is then approximated by the following formula:
$q\approx {\frac {\log(e_{\text{new}}/e_{\text{old}})}{\log(h_{\text{new}}/h_{\text{old}})}},$
which comes from writing the truncation error, at the old and new grid spacings, as
$e=|y_{n}-f(x_{n})|={\mathcal {O}}(h^{q}).$
The error $e$ is, more specifically, a global truncation error (GTE), in that it represents a sum of errors accumulated over all $n$ iterations, as opposed to a local truncation error (LTE) over just one iteration.
Example of discretization methods
Consider the ordinary differential equation
${\frac {dy}{dx}}=-\kappa y$
with initial condition $y(0)=y_{0}$. We can solve this equation using the Forward Euler scheme for numerical discretization:
${\frac {y_{n+1}-y_{n}}{h}}=-\kappa y_{n},$
which generates the sequence
$y_{n+1}=y_{n}(1-h\kappa ).$
In terms of $y(0)=y_{0}$, this sequence is as follows, from the Binomial theorem:
$y_{n}=y_{0}(1-h\kappa )^{n}=y_{0}\left(1-nh\kappa +n(n-1){\frac {h^{2}\kappa ^{2}}{2}}+....\right).$
The exact solution to this ODE is $y=f(x)=y_{0}\exp(-\kappa x)$, corresponding to the following Taylor expansion in $h\kappa $ for $h\kappa \ll 1$:
$f(x_{n})=f(nh)=y_{0}\exp(-\kappa nh)=y_{0}\left[\exp(-\kappa h)\right]^{n}=y_{0}\left(1-h\kappa +{\frac {h^{2}\kappa ^{2}}{2}}+....\right)^{n}=y_{0}\left(1-nh\kappa +{\frac {n^{2}h^{2}\kappa ^{2}}{2}}+...\right).$
In this case, the truncation error is
$e=|y_{n}-f(x_{n})|={\frac {nh^{2}\kappa ^{2}}{2}}={\mathcal {O}}(h^{2}),$
so $(y_{n})$ converges to $f(x_{n})$ with a convergence rate $q=2$.
Examples (continued)
The sequence $(d_{k})$ with $d_{k}=1/(k+1)$ was introduced above. This sequence converges with order 1 according to the convention for discretization methods.
The sequence $(a_{k})$ with $a_{k}=2^{-k}$, which was also introduced above, converges with order q for every number q. It is said to converge exponentially using the convention for discretization methods. However, it only converges linearly (that is, with order 1) using the convention for iterative methods.
Recurrent sequences and fixed points
The case of recurrent sequences $x_{n+1}:=f(x_{n})$ which occurs in dynamical systems and in the context of various fixed-point theorems is of particular interest. Assuming that the relevant derivatives of f are continuous, one can (easily) show that for a fixed point $f(p)=p$ such that $|f'(p)|<1$, one has at least linear convergence for any starting value $x_{0}$ sufficiently close to p. If $|f'(p)|=0$ and $|f''(p)|<1$, then one has at least quadratic convergence, and so on. If $|f'(p)|>1$, then one has a repulsive fixed point and no starting value will produce a sequence converging to p (unless one directly jumps to the point p itself).
Acceleration of convergence
Many methods exist to increase the rate of convergence of a given sequence, i.e. to transform a given sequence into one converging faster to the same limit. Such techniques are in general known as "series acceleration". The goal of the transformed sequence is to reduce the computational cost of the calculation. One example of series acceleration is Aitken's delta-squared process. These methods in general (and in particular Aitken's method) do not increase the order of convergence, and are useful only if initially the convergence is not faster than linear: If $(x_{n})$ convergences linearly, one gets a sequence $(a_{n})$ that still converges linearly (except for pathologically designed special cases), but faster in the sense that $\lim(a_{n}-L)/(x_{n}-L)=0$. On the other hand, if the convergence is already of order ≥ 2, Aitken's method will bring no improvement.
References
1. Ruye, Wang (2015-02-12). "Order and rate of convergence". hmc.edu. Retrieved 2020-07-31.
2. Senning, Jonathan R. "Computing and Estimating the Rate of Convergence" (PDF). gordon.edu. Retrieved 2020-08-07.
3. Bockelman, Brian (2005). "Rates of Convergence". math.unl.edu. Retrieved 2020-07-31.
4. Hundley, Douglas. "Rate of Convergence" (PDF). Whitman College. Retrieved 2020-12-13.{{cite web}}: CS1 maint: url-status (link)
5. Porta, F. A. (1989). "On Q-Order and R-Order of Convergence" (PDF). Journal of Optimization Theory and Applications. 63 (3): 415–431. doi:10.1007/BF00939805. S2CID 116192710. Retrieved 2020-07-31.
6. Senning, Jonathan R. "Computing and Estimating the Rate of Convergence" (PDF). gordon.edu. Retrieved 2020-08-07.
7. Arnold, Mark. "Order of Convergence" (PDF). University of Arkansas. Retrieved 2022-12-13.{{cite web}}: CS1 maint: url-status (link)
8. Van Tuyl, Andrew H. (1994). "Acceleration of convergence of a family of logarithmically convergent sequences" (PDF). Mathematics of Computation. 63 (207): 229–246. doi:10.2307/2153571. JSTOR 2153571. Retrieved 2020-08-02.
9. Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-30303-1.
Literature
The simple definition is used in
• Michelle Schatzman (2002), Numerical analysis: a mathematical introduction, Clarendon Press, Oxford. ISBN 0-19-850279-6.
The extended definition is used in
• Walter Gautschi (1997), Numerical analysis: an introduction, Birkhäuser, Boston. ISBN 0-8176-3895-4.
• Endre Süli and David Mayers (2003), An introduction to numerical analysis, Cambridge University Press. ISBN 0-521-00794-1.
The Big O definition is used in
• Richard L. Burden and J. Douglas Faires (2001), Numerical Analysis (7th ed.), Brooks/Cole. ISBN 0-534-38216-9
The terms Q-linear and R-linear are used in; The Big O definition when using Taylor series is used in
• Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin, New York: Springer-Verlag. pp. 619+620. ISBN 978-0-387-30303-1..
Differential equations
Classification
Operations
• Differential operator
• Notation for differentiation
• Ordinary
• Partial
• Differential-algebraic
• Integro-differential
• Fractional
• Linear
• Non-linear
• Holonomic
Attributes of variables
• Dependent and independent variables
• Homogeneous
• Nonhomogeneous
• Coupled
• Decoupled
• Order
• Degree
• Autonomous
• Exact differential equation
• On jet bundles
Relation to processes
• Difference (discrete analogue)
• Stochastic
• Stochastic partial
• Delay
Solutions
Existence/uniqueness
• Picard–Lindelöf theorem
• Peano existence theorem
• Carathéodory's existence theorem
• Cauchy–Kowalevski theorem
Solution topics
• Wronskian
• Phase portrait
• Phase space
• Lyapunov stability
• Asymptotic stability
• Exponential stability
• Rate of convergence
• Series solutions
• Integral solutions
• Numerical integration
• Dirac delta function
Solution methods
• Inspection
• Substitution
• Separation of variables
• Method of undetermined coefficients
• Variation of parameters
• Integrating factor
• Integral transforms
• Euler method
• Finite difference method
• Crank–Nicolson method
• Runge–Kutta methods
• Finite element method
• Finite volume method
• Galerkin method
• Perturbation theory
Applications
• List of named differential equations
Mathematicians
• Isaac Newton
• Gottfried Wilhelm Leibniz
• Leonhard Euler
• Jacob Bernoulli
• Émile Picard
• Józef Maria Hoene-Wroński
• Ernst Lindelöf
• Rudolf Lipschitz
• Joseph-Louis Lagrange
• Augustin-Louis Cauchy
• John Crank
• Phyllis Nicolson
• Carl David Tolmé Runge
• Martin Kutta
• Sofya Kovalevskaya
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Wikipedia
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Quadratic variation
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
Definition
Suppose that $X_{t}$ is a real-valued stochastic process defined on a probability space $(\Omega ,{\mathcal {F}},\mathbb {P} )$ and with time index $t$ ranging over the non-negative real numbers. Its quadratic variation is the process, written as $[X]_{t}$, defined as
$[X]_{t}=\lim _{\Vert P\Vert \rightarrow 0}\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}$
where $P$ ranges over partitions of the interval $[0,t]$ and the norm of the partition $P$ is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite 1-variation for every $t>0$ in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian motion.
More generally, the covariation (or cross-variance) of two processes $X$ and $Y$ is
$[X,Y]_{t}=\lim _{\Vert P\Vert \to 0}\sum _{k=1}^{n}\left(X_{t_{k}}-X_{t_{k-1}}\right)\left(Y_{t_{k}}-Y_{t_{k-1}}\right).$
The covariation may be written in terms of the quadratic variation by the polarization identity:
$[X,Y]_{t}={\tfrac {1}{2}}([X+Y]_{t}-[X]_{t}-[Y]_{t}).$
Notation: the quadratic variation is also notated as $\langle X\rangle _{t}$ or $\langle X,X\rangle _{t}$.
Finite variation processes
A process $X$ is said to have finite variation if it has bounded variation over every finite time interval (with probability 1). Such processes are very common including, in particular, all continuously differentiable functions. The quadratic variation exists for all continuous finite variation processes, and is zero.
This statement can be generalized to non-continuous processes. Any càdlàg finite variation process $X$ has quadratic variation equal to the sum of the squares of the jumps of $X$. To state this more precisely, the left limit of $X_{t}$ with respect to $t$ is denoted by $X_{t-}$, and the jump of $X$ at time $t$ can be written as $\Delta X_{t}=X_{t}-X_{t-}$. Then, the quadratic variation is given by
$[X]_{t}=\sum _{0<s\leq t}(\Delta X_{s})^{2}.$
The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here, $P$ is a partition of the interval $[0,t]$, and $V_{t}(X)$ is the variation of $X$ over $[0,t]$.
${\begin{aligned}\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}&\leq \max _{k\leq n}|X_{t_{k}}-X_{t_{k-1}}|\sum _{k=1}^{n}|X_{t_{k}}-X_{t_{k-1}}|\\&\leq \max _{|u-v|\leq \Vert P\Vert }|X_{u}-X_{v}|V_{t}(X).\end{aligned}}$
By the continuity of $X$, this vanishes in the limit as $\Vert P\Vert $ goes to zero.
Itô processes
The quadratic variation of a standard Brownian motion $B$ exists, and is given by $[B]_{t}=t$, however the limit in the definition is meant in the $L^{2}$ sense and not pathwise. This generalizes to Itô processes that, by definition, can be expressed in terms of Itô integrals
${\begin{aligned}X_{t}&=X_{0}+\int _{0}^{t}\sigma _{s}\,dB_{s}+\int _{0}^{t}\mu _{s}\,d[B]_{s}\\&=X_{0}+\int _{0}^{t}\sigma _{s}\,dB_{s}+\int _{0}^{t}\mu _{s}\,ds,\end{aligned}}$
where $B$ is a Brownian motion. Any such process has quadratic variation given by
$[X]_{t}=\int _{0}^{t}\sigma _{s}^{2}\,ds.$
Semimartingales
Quadratic variations and covariations of all semimartingales can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in Itô's lemma, which is the generalization of the chain rule to the Itô integral. The quadratic covariation also appears in the integration by parts formula
$X_{t}Y_{t}=X_{0}Y_{0}+\int _{0}^{t}X_{s-}\,dY_{s}+\int _{0}^{t}Y_{s-}\,dX_{s}+[X,Y]_{t},$
which can be used to compute $[X,Y]$.
Alternatively this can be written as a stochastic differential equation:
$\,d(X_{t}Y_{t})=X_{t-}\,dY_{t}+Y_{t-}\,dX_{t}+\,dX_{t}\,dY_{t},$
where $\,dX_{t}\,dY_{t}=\,d[X,Y]_{t}.$
Martingales
All càdlàg martingales, and local martingales have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales. It can be shown that the quadratic variation $[M]$ of a general locally square integrable martingale $M$ is the unique right-continuous and increasing process starting at zero, with jumps $\Delta [M]=\Delta M^{2}$ and such that $M^{2}-[M]$ is a local martingale. A proof of existence of $M$ (without using stochastic calculus) is given in Karandikar–Rao (2014).
A useful result for square integrable martingales is the Itô isometry, which can be used to calculate the variance of Itô integrals,
$\operatorname {E} \left(\left(\int _{0}^{t}H\,dM\right)^{2}\right)=\operatorname {E} \left(\int _{0}^{t}H^{2}\,d[M]\right).$
This result holds whenever $M$ is a càdlàg square integrable martingale and $H$ is a bounded predictable process, and is often used in the construction of the Itô integral.
Another important result is the Burkholder–Davis–Gundy inequality. This gives bounds for the maximum of a martingale in terms of the quadratic variation. For a local martingale $M$ starting at zero, with maximum denoted by $M_{t}*=\operatorname {sup} _{s\in [0,t]}|M_{s}|$, and any real number $p\geq 1$, the inequality is
$c_{p}\operatorname {E} ([M]_{t}^{p/2})\leq \operatorname {E} ((M_{t}^{*})^{p})\leq C_{p}\operatorname {E} ([M]_{t}^{p/2}).$
Here, $c_{p}<C_{p}$ are constants depending on the choice of $p$, but not depending on the martingale $M$ or time $t$ used. If $M$ is a continuous local martingale, then the Burkholder–Davis–Gundy inequality holds for any $p>0$.
An alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales. This is written as $\langle M_{t}\rangle $, and is defined to be the unique right-continuous and increasing predictable process starting at zero such that $M^{2}-\langle M\rangle $ is a local martingale. Its existence follows from the Doob–Meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation.
See also
• Total variation
• Bounded variation
References
• Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 978-3-540-00313-7
• Karandikar, Rajeeva L.; Rao, B. V. (2014). "On quadratic variation of martingales". Proceedings - Mathematical Sciences. 124 (3): 457–469. doi:10.1007/s12044-014-0179-2. S2CID 120031445.
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Wikipedia
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Quadratic differential
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural interpretation as the cotangent space to the Riemann moduli space, or Teichmüller space.
Local form
Each quadratic differential on a domain $U$ in the complex plane may be written as $f(z)\,dz\otimes dz$, where $z$ is the complex variable, and $f$ is a complex-valued function on $U$. Such a "local" quadratic differential is holomorphic if and only if $f$ is holomorphic. Given a chart $\mu $ for a general Riemann surface $R$ and a quadratic differential $q$ on $R$, the pull-back $(\mu ^{-1})^{*}(q)$ defines a quadratic differential on a domain in the complex plane.
Relation to abelian differentials
If $\omega $ is an abelian differential on a Riemann surface, then $\omega \otimes \omega $ is a quadratic differential.
Singular Euclidean structure
A holomorphic quadratic differential $q$ determines a Riemannian metric $|q|$ on the complement of its zeroes. If $q$ is defined on a domain in the complex plane, and $q=f(z)\,dz\otimes dz$, then the associated Riemannian metric is $|f(z)|(dx^{2}+dy^{2})$, where $z=x+iy$. Since $f$ is holomorphic, the curvature of this metric is zero. Thus, a holomorphic quadratic differential defines a flat metric on the complement of the set of $z$ such that $f(z)=0$.
References
• Kurt Strebel, Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 5. Springer-Verlag, Berlin, 1984. xii + 184 pp. ISBN 3-540-13035-7.
• Y. Imayoshi and M. Taniguchi, M. An introduction to Teichmüller spaces. Translated and revised from the Japanese version by the authors. Springer-Verlag, Tokyo, 1992. xiv + 279 pp. ISBN 4-431-70088-9.
• Frederick P. Gardiner, Teichmüller Theory and Quadratic Differentials. Wiley-Interscience, New York, 1987. xvii + 236 pp. ISBN 0-471-84539-6.
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Wikipedia
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Quadratic eigenvalue problem
In mathematics, the quadratic eigenvalue problem[1] (QEP), is to find scalar eigenvalues $\lambda $, left eigenvectors $y$ and right eigenvectors $x$ such that
$Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,$
where $Q(\lambda )=\lambda ^{2}M+\lambda C+K$, with matrix coefficients $M,\,C,K\in \mathbb {C} ^{n\times n}$ and we require that $M\,\neq 0$, (so that we have a nonzero leading coefficient). There are $2n$ eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. $Q(\lambda )$ is also known as a quadratic polynomial matrix.
Spectral theory
A QEP is said to be regular if ${\text{Det}}(Q(\lambda ))\not \equiv 0$ identically. The coefficient of the $\lambda ^{2n}$ term in ${\text{Det}}(Q(\lambda ))$ is ${\text{Det}}(M)$, implying that the QEP is regular if $M$ is nonsingular.
Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, $\lambda ^{2}Q(\lambda ^{-1})=\lambda ^{2}K+\lambda C+M$. As there are $2n$ eigenvectors in a $n$ dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.
Applications
Systems of differential equations
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:
$Mq''(t)+Cq'(t)+Kq(t)=0$
Where $q(t)\in \mathbb {R} ^{n}$, and $M,C,K\in \mathbb {R} ^{n\times n}$. If all quadratic eigenvalues of $Q(\lambda )=\lambda ^{2}M+\lambda C+K$ are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
$q(t)=\sum _{j=1}^{2n}\alpha _{j}x_{j}e^{\lambda _{j}t}=Xe^{\Lambda t}\alpha $
Where $\Lambda ={\text{Diag}}([\lambda _{1},\ldots ,\lambda _{2n}])\in \mathbb {R} ^{2n\times 2n}$ are the quadratic eigenvalues, $X=[x_{1},\ldots ,x_{2n}]\in \mathbb {R} ^{n\times 2n}$ are the $2n$ right quadratic eigenvectors, and $\alpha =[\alpha _{1},\cdots ,\alpha _{2n}]^{\top }\in \mathbb {R} ^{2n}$ is a parameter vector determined from the initial conditions on $q$ and $q'$. Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.
Finite element methods
A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, $Q(\lambda )$ has the form $Q(\lambda )=\lambda ^{2}M+\lambda C+K$, where $M$ is the mass matrix, $C$ is the damping matrix and $K$ is the stiffness matrix. Other applications include vibro-acoustics and fluid dynamics.
Methods of solution
Direct methods for solving the standard or generalized eigenvalue problems $Ax=\lambda x$ and $Ax=\lambda Bx$ are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials. One approach is to transform the quadratic matrix polynomial to a linear matrix pencil ($A-\lambda B$), and solve a generalized eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
The most common linearization is the first companion linearization
$L1(\lambda )={\begin{bmatrix}0&N\\-K&-C\end{bmatrix}}-\lambda {\begin{bmatrix}N&0\\0&M\end{bmatrix}},$
with corresponding eigenvector
$z={\begin{bmatrix}x\\\lambda x\end{bmatrix}}.$
For convenience, one often takes $N$ to be the $n\times n$ identity matrix. We solve $L(\lambda )z=0$ for $\lambda $ and $z$, for example by computing the Generalized Schur form. We can then take the first $n$ components of $z$ as the eigenvector $x$ of the original quadratic $Q(\lambda )$.
Another common linearization is given by
$L2(\lambda )={\begin{bmatrix}-K&0\\0&N\end{bmatrix}}-\lambda {\begin{bmatrix}C&M\\N&0\end{bmatrix}}.$
In the case when either $A$ or $B$ is a Hamiltonian matrix and the other is a skew-Hamiltonian matrix, the following linearizations can be used.
$L3(\lambda )={\begin{bmatrix}K&0\\C&K\end{bmatrix}}-\lambda {\begin{bmatrix}0&K\\-M&0\end{bmatrix}}.$
$L4(\lambda )={\begin{bmatrix}0&-K\\M&0\end{bmatrix}}-\lambda {\begin{bmatrix}M&C\\0&M\end{bmatrix}}.$
References
1. F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286.
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Wikipedia
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Quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others.
Not to be confused with quadratic function or quadratic equation.
Given a general quadratic equation of the form
$ax^{2}+bx+c=0$
whose discriminant $b^{2}-4ac$ is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is:
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$
where the plus–minus symbol "±" indicates that the quadratic equation has two solutions.[1] Written separately, they become:
${\begin{aligned}x_{1}&={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\\x_{2}&={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}$
Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the x-values at which any parabola, explicitly given as y = ax2 + bx + c, crosses the x-axis.[2]
As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola,[3] and the number of real zeros the quadratic equation contains.[4]
The expression b2 − 4ac is known as the discriminant. If a, b, and c are real numbers and a ≠ 0 then
1. When b2 − 4ac > 0, there are two distinct real roots or solutions to the equation ax2 + bx + c = 0.
2. When b2 − 4ac = 0, there is one repeated real solution.
3. When b2 − 4ac < 0, there are two distinct complex solutions, which are complex conjugates of each other.
Equivalent formulations
The quadratic formula, in the case when the discriminant $b^{2}-4ac$ is positive, may also be written as
$x=-{\frac {b}{2a}}\pm {\sqrt {\frac {b^{2}-4ac}{4a^{2}}}}\,,$
which may be simplified to
$x=-{\frac {b}{2a}}\pm {\sqrt {\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}}}\,.$
This version of the formula makes it easy to find the roots when using a calculator.
When b is an even integer, it is usually easier to use the reduced formula
$x={\frac {-{\frac {b}{2}}\pm {\sqrt {\left({\frac {b}{2}}\right)^{2}-ac}}}{a}}$
In the case when the discriminant $b^{2}-4ac$ is negative, complex roots are involved. The quadratic formula can be written as:
$x=-{\frac {b}{2a}}\pm i{\sqrt {\left|\left({\frac {b}{2a}}\right)^{2}-{\frac {c}{a}}\right|}}\,.$
Muller's method
A lesser known quadratic formula, also named "citardauq", which is used in Muller's method and which can be found from Vieta's formulas, provides (assuming a ≠ 0, c ≠ 0) the same roots via the equation:
$x={\frac {-2c}{b\pm {\sqrt {b^{2}-4ac}}}}={\frac {2c}{-b\mp {\sqrt {b^{2}-4ac}}}}.$
For positive $b$, the subtraction causes cancellation in the standard formula (respectively negative $b$ and addition), resulting in poor accuracy. In this case, switching to Muller's formula with the opposite sign is a good workaround.
Formulations based on alternative parameterizations
The standard parameterization of the quadratic equation is
$ax^{2}+bx+c=0\,.$
Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as
$ax^{2}-2b_{1}x+c=0,$
where $b_{1}=-b/2$,[5]
or
$ax^{2}+2b_{2}x+c=0,$
where $b_{2}=b/2$.[6]
These alternative parameterizations result in slightly different forms for the solution, but they are otherwise equivalent to the standard parameterization.
Derivations of the formula
Many different methods to derive the quadratic formula are available in the literature. The standard one is a simple application of the completing the square technique.[7][8][9][10] Alternative methods are sometimes simpler than completing the square, and may offer interesting insight into other areas of mathematics.
By completing the square
Divide the quadratic equation by $a$, which is allowed because $a$ is non-zero:
$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=0\,.$
Subtract c/a from both sides of the equation, yielding:
$x^{2}+{\frac {b}{a}}x=-{\frac {c}{a}}\,.$
The quadratic equation is now in a form to which the method of completing the square is applicable. In fact, by adding a constant to both sides of the equation such that the left hand side becomes a complete square, the quadratic equation becomes:
$x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+\left({\frac {b}{2a}}\right)^{2}\,,$
which produces:
$\left(x+{\frac {b}{2a}}\right)^{2}=-{\frac {c}{a}}+{\frac {b^{2}}{4a^{2}}}\,.$
Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:
$\left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}\,.$
The square has thus been completed. If the discriminant $b^{2}-4ac$ is positive, we can take the square root of both sides, yielding the following equation:
$x+{\frac {b}{2a}}=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}\,.$
(In fact, this equation remains true even if the discriminant is not positive, by interpreting the root of the discriminant as any of its two opposite complex roots.)
In which case, isolating the $x$ would give the quadratic formula:
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\,.$
There are many alternatives of this derivation with minor differences, mostly concerning the manipulation of $a$.
Shorter method
Completing the square can also be accomplished by a sometimes shorter and simpler sequence:[11]
1. Multiply each side by $4a$,
2. Rearrange.
3. Add $b^{2}$ to both sides to complete the square.
4. The left side is the outcome of the polynomial $(2ax+b)^{2}$.
5. Take the square root of both sides.
6. Isolate $x$.
In which case, the quadratic formula can also be derived as follows:
${\begin{array}{llllllc}ax^{2}&+&bx&+&c&=&0\\4a^{2}x^{2}&+&4abx&+&4ac&=&0\\4a^{2}x^{2}&+&4abx&&&=&-4ac\\4a^{2}x^{2}&+&4abx&+&b^{2}&=&b^{2}-4ac\\&&(2ax+b)^{2}&&&=&b^{2}-4ac\\[1ex]{\text{ (valid if }}b^{2}-4ac{\text{ is positive)}}&&2ax+b&&&=&\pm {\sqrt {b^{2}-4ac}}\\[3ex]&&2ax&&&=&-b\pm {\sqrt {b^{2}-4ac}}\\[1ex]&&&&x&=&{\dfrac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\,.\\[-1ex]\,\end{array}}$
This derivation of the quadratic formula is ancient and was known in India at least as far back as 1025.[12] Compared with the derivation in standard usage, this alternate derivation avoids fractions and squared fractions until the last step and hence does not require a rearrangement after step 3 to obtain a common denominator in the right side.[11]
By substitution
Another technique is solution by substitution.[13] In this technique, we substitute $x=y+m$ into the quadratic to get:
$a(y+m)^{2}+b(y+m)+c=0\,.$
Expanding the result and then collecting the powers of $y$ produces:
$ay^{2}+y(2am+b)+\left(am^{2}+bm+c\right)=0\,.$
We have not yet imposed a second condition on $y$ and $m$, so we now choose $m$ so that the middle term vanishes. That is, $2am+b=0$ or $\textstyle m={\frac {-b}{2a}}$.
$ay^{2}+y(\ \ \ 0\ \ )+\left(am^{2}+bm+c\right)=0\,.$
$ay^{2}+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(am^{2}+bm+c\right)=0\,.$
Subtracting the constant term from both sides of the equation (to move it to the right hand side) and then dividing by $a$ gives:
$y^{2}={\frac {-\left(am^{2}+bm+c\right)}{a}}\,.$
Substituting for $m$ gives:
$y^{2}={\frac {-\left({\frac {b^{2}}{4a}}+{\frac {-b^{2}}{2a}}+c\right)}{a}}={\frac {b^{2}-4ac}{4a^{2}}}\,.$
Therefore,
$y=\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}$
By re-expressing $y$ in terms of $x$ using the formula $ x=y+m=y-{\frac {b}{2a}}$ , the usual quadratic formula can then be obtained:
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\,.$
By using algebraic identities
The following method was used by many historical mathematicians:[14]
Let the roots of the standard quadratic equation be r1 and r2. The derivation starts by recalling the identity:
$(r_{1}-r_{2})^{2}=(r_{1}+r_{2})^{2}-4r_{1}r_{2}\,.$
Taking the square root on both sides, we get:
$r_{1}-r_{2}=\pm {\sqrt {(r_{1}+r_{2})^{2}-4r_{1}r_{2}}}\,.$
Since the coefficient a ≠ 0, we can divide the standard equation by a to obtain a quadratic polynomial having the same roots. Namely,
$x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}=(x-r_{1})(x-r_{2})=x^{2}-(r_{1}+r_{2})x+r_{1}r_{2}\,.$
From this we can see that the sum of the roots of the standard quadratic equation is given by −b/a, and the product of those roots is given by c/a. Hence the identity can be rewritten as:
$r_{1}-r_{2}=\pm {\sqrt {\left(-{\frac {b}{a}}\right)^{2}-4{\frac {c}{a}}}}=\pm {\sqrt {{\frac {b^{2}}{a^{2}}}-{\frac {4ac}{a^{2}}}}}=\pm {\frac {\sqrt {b^{2}-4ac}}{a}}\,.$
Now,
$r_{1}={\frac {(r_{1}+r_{2})+(r_{1}-r_{2})}{2}}={\frac {-{\frac {b}{a}}\pm {\frac {\sqrt {b^{2}-4ac}}{a}}}{2}}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\,.$
Since r2 = −r1 − b/a, if we take
$r_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}$
then we obtain
$r_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\,;$
and if we instead take
$r_{1}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}$
then we calculate that
$r_{2}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\,.$
Combining these results by using the standard shorthand ±, we have that the solutions of the quadratic equation are given by:
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\,.$
By Lagrange resolvents
Further information: Lagrange resolvents
An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents,[15] which is an early part of Galois theory.[16] This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group.
This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial
$x^{2}+px+q\,,$
assume that it factors as
$x^{2}+px+q=(x-\alpha )(x-\beta )\,,$
Expanding yields
$x^{2}+px+q=x^{2}-(\alpha +\beta )x+\alpha \beta \,,$
where p = −(α + β) and q = αβ.
Since the order of multiplication does not matter, one can switch α and β and the values of p and q will not change: one can say that p and q are symmetric polynomials in α and β. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can be expressed in terms of α + β and αβ. The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree n is related to the ways of rearranging ("permuting") n terms, which is called the symmetric group on n letters, and denoted Sn. For the quadratic polynomial, the only ways to rearrange two terms is to leave them be or to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.
To find the roots α and β, consider their sum and difference:
${\begin{aligned}r_{1}&=\alpha +\beta \\r_{2}&=\alpha -\beta \,.\end{aligned}}$
These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:
${\begin{aligned}\alpha &=\textstyle {\frac {1}{2}}\left(r_{1}+r_{2}\right)\\\beta &=\textstyle {\frac {1}{2}}\left(r_{1}-r_{2}\right)\,.\end{aligned}}$
Thus, solving for the resolvents gives the original roots.
Now r1 = α + β is a symmetric function in α and β, so it can be expressed in terms of p and q, and in fact r1 = −p as noted above. But r2 = α − β is not symmetric, since switching α and β yields −r2 = β − α (formally, this is termed a group action of the symmetric group of the roots). Since r2 is not symmetric, it cannot be expressed in terms of the coefficients p and q, as these are symmetric in the roots and thus so is any polynomial expression involving them. Changing the order of the roots only changes r2 by a factor of −1, and thus the square r22 = (α − β)2 is symmetric in the roots, and thus expressible in terms of p and q. Using the equation
$(\alpha -\beta )^{2}=(\alpha +\beta )^{2}-4\alpha \beta $
yields
$r_{2}^{2}=p^{2}-4q$
and thus
$r_{2}=\pm {\sqrt {p^{2}-4q}}$
If one takes the positive root, breaking symmetry, one obtains:
${\begin{aligned}r_{1}&=-p\\r_{2}&={\sqrt {p^{2}-4q}}\end{aligned}}$
and thus
${\begin{aligned}\alpha &={\tfrac {1}{2}}\left(-p+{\sqrt {p^{2}-4q}}\right)\\\beta &={\tfrac {1}{2}}\left(-p-{\sqrt {p^{2}-4q}}\right)\,.\end{aligned}}$
Thus the roots are
${\tfrac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\right)$
which is the quadratic formula. Substituting p = b/a, q = c/a yields the usual form for when a quadratic is not monic. The resolvents can be recognized as r1/2 = −p/2 = −b/2a being the vertex, and r22 = p2 − 4q is the discriminant (of a monic polynomial).
A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating r2 and r3, which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved.[15] The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and, in fact, solutions to quintic equations in general cannot be expressed using only roots.
Historical development
The earliest methods for solving quadratic equations were geometric. Babylonian cuneiform tablets contain problems reducible to solving quadratic equations.[17]: 34 The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.[18]
The Greek mathematician Euclid (circa 300 BC) used geometric methods to solve quadratic equations in Book 2 of his Elements, an influential mathematical treatise.[17]: 39 Rules for quadratic equations appear in the Chinese The Nine Chapters on the Mathematical Art circa 200 BC.[19][20] In his work Arithmetica, the Greek mathematician Diophantus (circa 250 AD) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid.[17]: 39 His solution gives only one root, even when both roots are positive.[21]
The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD,[22] but written in words instead of symbols.[23] His solution of the quadratic equation ax2 + bx = c was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."[24] This is equivalent to:
$x={\frac {{\sqrt {c\cdot 4a+b^{2}}}-b}{2a}}\,.$
Śrīdharācāryya (870–930 AD), an Indian mathematician also came up with a similar algorithm for solving quadratic equations, though there is no indication that he considered both the roots.[25] The 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī solved quadratic equations algebraically.[17]: 42 The quadratic formula covering all cases was first obtained by Simon Stevin in 1594.[26] In 1637 René Descartes published La Géométrie containing special cases of the quadratic formula in the form we know today.[27]
Significant uses
Geometric significance
In terms of coordinate geometry, a parabola is a curve whose (x, y)-coordinates are described by a second-degree polynomial, i.e. any equation of the form:
$y=p(x)=a_{2}x^{2}+a_{1}x+a_{0}\,,$
where p represents the polynomial of degree 2 and a0, a1, and a2 ≠ 0 are constant coefficients whose subscripts correspond to their respective term's degree. The geometrical interpretation of the quadratic formula is that it defines the points on the x-axis where the parabola will cross the axis. Additionally, if the quadratic formula was looked at as two terms,
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}=-{\frac {b}{2a}}\pm {\frac {\sqrt {b^{2}-4ac}}{2a}}$
the axis of symmetry appears as the line x = −b/2a. The other term, √b2 − 4ac/2a, gives the distance the zeros are away from the axis of symmetry, where the plus sign represents the distance to the right, and the minus sign represents the distance to the left.
If this distance term were to decrease to zero, the value of the axis of symmetry would be the x value of the only zero, that is, there is only one possible solution to the quadratic equation. Algebraically, this means that √b2 − 4ac = 0, or simply b2 − 4ac = 0 (where the left-hand side is referred to as the discriminant). This is one of three cases, where the discriminant indicates how many zeros the parabola will have. If the discriminant is positive, the distance would be non-zero, and there will be two solutions. However, there is also the case where the discriminant is less than zero, and this indicates the distance will be imaginary – or some multiple of the complex unit i, where i = √−1 – and the parabola's zeros will be complex numbers. The complex roots will be complex conjugates, where the real part of the complex roots will be the value of the axis of symmetry. There will be no real values of x where the parabola crosses the x-axis.
Dimensional analysis
If the constants a, b, and/or c are not unitless then the units of x must be equal to the units of b/a, due to the requirement that ax2 and bx agree on their units. Furthermore, by the same logic, the units of c must be equal to the units of b2/a, which can be verified without solving for x. This can be a powerful tool for verifying that a quadratic expression of physical quantities has been set up correctly.
See also
• Fundamental theorem of algebra
• Vieta's formulas
References
1. Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, p. 219, ISBN 978-0-470-55964-2
2. "Understanding the quadratic formula". Khan Academy. Retrieved 2019-11-10.
3. "Axis of Symmetry of a Parabola. How to find axis from equation or from a graph. To find the axis of symmetry ..." www.mathwarehouse.com. Retrieved 2019-11-10.
4. "Discriminant review". Khan Academy. Retrieved 2019-11-10.
5. Kahan, Willian (November 20, 2004), On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic (PDF), retrieved 2012-12-25
6. "Quadratic Formula", Proof Wiki, retrieved 2016-10-08
7. Rich, Barnett; Schmidt, Philip (2004), Schaum's Outline of Theory and Problems of Elementary Algebra, The McGraw–Hill Companies, Chapter 13 §4.4, p. 291, ISBN 0-07-141083-X
8. Li, Xuhui. An Investigation of Secondary School Algebra Teachers' Mathematical Knowledge for Teaching Algebraic Equation Solving, p. 56 (ProQuest, 2007): "The quadratic formula is the most general method for solving quadratic equations and is derived from another general method: completing the square."
9. Rockswold, Gary. College algebra and trigonometry and precalculus, p. 178 (Addison Wesley, 2002).
10. Beckenbach, Edwin et al. Modern college algebra and trigonometry, p. 81 (Wadsworth Pub. Co., 1986).
11. Hoehn, Larry (1975). "A More Elegant Method of Deriving the Quadratic Formula". The Mathematics Teacher. 68 (5): 442–443. doi:10.5951/MT.68.5.0442.
12. Smith, David E. (1958). History of Mathematics, Vol. II. Dover Publications. p. 446. ISBN 0486204308.
13. Joseph J. Rotman. (2010). Advanced modern algebra (Vol. 114). American Mathematical Soc. Section 1.1
14. Debnath, Lokenath (2009). "The legacy of Leonhard Euler – a tricentennial tribute". International Journal of Mathematical Education in Science and Technology. 40 (3): 353–388. doi:10.1080/00207390802642237. S2CID 123048345.
15. Clark, A. (1984). Elements of abstract algebra. Courier Corporation. p. 146.
16. Prasolov, Viktor; Solovyev, Yuri (1997), Elliptic functions and elliptic integrals, AMS Bookstore, p. 134, ISBN 978-0-8218-0587-9
17. Irving, Ron (2013). Beyond the Quadratic Formula. MAA. ISBN 978-0-88385-783-0.
18. The Cambridge Ancient History Part 2 Early History of the Middle East. Cambridge University Press. 1971. p. 530. ISBN 978-0-521-07791-0.
19. Aitken, Wayne. "A Chinese Classic: The Nine Chapters" (PDF). Mathematics Department, California State University. Retrieved 28 April 2013.
20. Smith, David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 380. ISBN 978-0-486-20430-7.
21. Smith, David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 134. ISBN 0-486-20429-4.
22. Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).
23. Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).
24. Stillwell, John (2004). Mathematics and Its History (2nd ed.). Springer. p. 87. ISBN 0-387-95336-1.
25. Sridhara-MacTutor
26. Struik, D. J.; Stevin, Simon (1958), The Principal Works of Simon Stevin, Mathematics (PDF), vol. II–B, C. V. Swets & Zeitlinger, p. 470
27. Rene Descartes. The Geometry.
Polynomials and polynomial functions
By degree
• Zero polynomial (degree undefined or −1 or −∞)
• Constant function (0)
• Linear function (1)
• Linear equation
• Quadratic function (2)
• Quadratic equation
• Cubic function (3)
• Cubic equation
• Quartic function (4)
• Quartic equation
• Quintic function (5)
• Sextic equation (6)
• Septic equation (7)
By properties
• Univariate
• Bivariate
• Multivariate
• Monomial
• Binomial
• Trinomial
• Irreducible
• Square-free
• Homogeneous
• Quasi-homogeneous
Tools and algorithms
• Factorization
• Greatest common divisor
• Division
• Horner's method of evaluation
• Resultant
• Discriminant
• Gröbner basis
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Wikipedia
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Quadratic growth
In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta notation, $f(x)=\Theta (x^{2})$.[1] This can be defined both continuously (for a real-valued function of a real variable) or discretely (for a sequence of real numbers, i.e., real-valued function of an integer or natural number variable).
Examples
Examples of quadratic growth include:
• Any quadratic polynomial.
• Certain integer sequences such as the triangular numbers. The $n$th triangular number has value $n(n+1)/2$, approximately $n^{2}/2$.
For a real function of a real variable, quadratic growth is equivalent to the second derivative being constant (i.e., the third derivative being zero), and thus functions with quadratic growth are exactly the quadratic polynomials, as these are the kernel of the third derivative operator $D^{3}$. Similarly, for a sequence (a real function of an integer or natural number variable), quadratic growth is equivalent to the second finite difference being constant (the third finite difference being zero),[2] and thus a sequence with quadratic growth is also a quadratic polynomial. Indeed, an integer-valued sequence with quadratic growth is a polynomial in the zeroth, first, and second binomial coefficient with integer values. The coefficients can be determined by taking the Taylor polynomial (if continuous) or Newton polynomial (if discrete).
Algorithmic examples include:
• The amount of time taken in the worst case by certain algorithms, such as insertion sort, as a function of the input length.[3]
• The numbers of live cells in space-filling cellular automaton patterns such as the breeder, as a function of the number of time steps for which the pattern is simulated.[4]
• Metcalfe's law stating that the value of a communications network grows quadratically as a function of its number of users.[5]
See also
• Exponential growth
References
1. Moore, Cristopher; Mertens, Stephan (2011), The Nature of Computation, Oxford University Press, p. 22, ISBN 9780191620805.
2. Kalman, Dan (1997), Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos, Cambridge University Press, p. 81, ISBN 9780883857076.
3. Estivill-Castro, Vladimir (1999), "Sorting and order statistics", in Atallah, Mikhail J. (ed.), Algorithms and Theory of Computation Handbook, Boca Raton, Florida: CRC, pp. 3-1–3-25, MR 1797171.
4. Griffeath, David; Hickerson, Dean (2003), "A two-dimensional cellular automaton crystal with irrational density", New constructions in cellular automata, St. Fe Inst. Stud. Sci. Complex., New York: Oxford Univ. Press, pp. 79–91, MR 2079729. See in particular p. 81: "A breeder is any pattern which grows quadratically by creating a steady stream of copies of a second object, each of which creates a stream of a third."
5. Rohlfs, Jeffrey H. (2003), "3.3 Metcalfe's law", Bandwagon Effects in High-technology Industries, MIT Press, pp. 29–30, ISBN 9780262681384.
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Wikipedia
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Quadratic integer
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form
x2 + bx + c = 0
with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers.
Common examples of quadratic integers are the square roots of rational integers, such as √2, and the complex number i = √−1, which generates the Gaussian integers. Another common example is the non-real cubic root of unity −1 + √−3/2, which generates the Eisenstein integers.
Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
History
Medieval Indian mathematicians had already discovered a multiplication of quadratic integers of the same D, which allowed them to solve some cases of Pell's equation.
The characterization given in § Explicit representation of the quadratic integers was first given by Richard Dedekind in 1871.[1][2]
Definition
A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number $x={\frac {-b\pm {\sqrt {b^{2}-4c}}}{2}}$, which solves an equation of the form x2 + bx + c = 0, with b and c integers. Each quadratic integer that is not an integer is not rational—namely, it's a real irrational number if b2 − 4c > 0 and non-real if b2 − 4c < 0—and lies in a uniquely determined quadratic field $\mathbb {Q} ({\sqrt {D}}\,)$, the extension of $\mathbb {Q} $ generated by the square root of the unique square-free integer D that satisfies b2 − 4c = De2 for some integer e. If D is positive, the quadratic integer is real. If D < 0, it is imaginary (that is, complex and non-real).
The quadratic integers (including the ordinary integers) that belong to a quadratic field $\mathbb {Q} ({\sqrt {D}}\,)$ form an integral domain called the ring of integers of $\mathbb {Q} ({\sqrt {D}}\,).$
Although the quadratic integers belonging to a given quadratic field form a ring, the set of all quadratic integers is not a ring because it is not closed under addition or multiplication. For example, $1+{\sqrt {2}}$ and ${\sqrt {3}}$ are quadratic integers, but $1+{\sqrt {2}}+{\sqrt {3}}$ and $(1+{\sqrt {2}})\cdot {\sqrt {3}}$ are not, as their minimal polynomials have degree four.
Explicit representation
Here and in the following, the quadratic integers that are considered belong to a quadratic field $\mathbb {Q} ({\sqrt {D}}\,),$ where D is a square-free integer. This does not restrict the generality, as the equality √a2D = a √D (for any positive integer a) implies $\mathbb {Q} ({\sqrt {D}}\,)=\mathbb {Q} ({\sqrt {a^{2}D}}\,).$
An element x of $\mathbb {Q} ({\sqrt {D}}\,)$ is a quadratic integer if and only if there are two integers a and b such that either
$x=a+b{\sqrt {D}},$
or, if D − 1 is a multiple of 4
$x={\frac {a}{2}}+{\frac {b}{2}}{\sqrt {D}},$ with a and b both odd
In other words, every quadratic integer may be written a + ωb , where a and b are integers, and where ω is defined by
$\omega ={\begin{cases}{\sqrt {D}}&{\mbox{if }}D\equiv 2,3{\pmod {4}}\\{{1+{\sqrt {D}}} \over 2}&{\mbox{if }}D\equiv 1{\pmod {4}}\end{cases}}$
(as D has been supposed square-free the case $D\equiv 0{\pmod {4}}$ is impossible, since it would imply that D is divisible by the square 4).[3]
Norm and conjugation
A quadratic integer in $\mathbb {Q} ({\sqrt {D}}\,)$ may be written
a + b√D,
where a and b are either both integers, or, only if D ≡ 1 (mod 4), both halves of odd integers. The norm of such a quadratic integer is
N (a + b√D ) = a2 − Db2.
The norm of a quadratic integer is always an integer. If D < 0, the norm of a quadratic integer is the square of its absolute value as a complex number (this is false if D > 0). The norm is a completely multiplicative function, which means that the norm of a product of quadratic integers is always the product of their norms.
Every quadratic integer a + b√D has a conjugate
${\overline {a+b{\sqrt {D}}}}=a-b{\sqrt {D}}.$
A quadratic integer has the same norm as its conjugate, and this norm is the product of the quadratic integer and its conjugate. The conjugate of a sum or a product of quadratic integers is the sum or the product (respectively) of the conjugates. This means that the conjugation is an automorphism of the ring of the integers of $\mathbb {Q} ({\sqrt {D}}\,)$—see § Quadratic integer rings, below.
Quadratic integer rings
Every square-free integer (different from 0 and 1) D defines a quadratic integer ring, which is the integral domain consisting of the algebraic integers contained in $\mathbf {Q} ({\sqrt {D}}\,).$ It is the set Z[ω] = {a + ωb : a, b ∈ Z}, where $\omega ={\tfrac {1+{\sqrt {D}}}{2}}$ if D = 4k + 1, and ω = √D otherwise. It is often denoted ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}$, because it is the ring of integers of $\mathbf {Q} ({\sqrt {D}}\,)$, which is the integral closure of Z in $\mathbf {Q} ({\sqrt {D}}\,).$ The ring Z[ω] consists of all roots of all equations x2 + Bx + C = 0 whose discriminant B2 − 4C is the product of D by the square of an integer. In particular √D belongs to Z[ω], being a root of the equation x2 − D = 0, which has 4D as its discriminant.
The square root of any integer is a quadratic integer, as every integer can be written n = m2D, where D is a square-free integer, and its square root is a root of x2 − m2D = 0.
The fundamental theorem of arithmetic is not true in many rings of quadratic integers. However, there is a unique factorization for ideals, which is expressed by the fact that every ring of algebraic integers is a Dedekind domain. Being the simplest examples of algebraic integers, quadratic integers are commonly the starting examples of most studies of algebraic number theory.[4]
The quadratic integer rings divide in two classes depending on the sign of D. If D > 0, all elements of ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}$ are real, and the ring is a real quadratic integer ring. If D < 0, the only real elements of ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}$ are the ordinary integers, and the ring is a complex quadratic integer ring.
For real quadratic integer rings, the class number – which measures the failure of unique factorization – is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924.
Units
A quadratic integer is a unit in the ring of the integers of $\mathbb {Q} ({\sqrt {D}}\,)$ if and only if its norm is 1 or −1. In the first case its multiplicative inverse is its conjugate. It is the negation of its conjugate in the second case.
If D < 0, the ring of the integers of $\mathbb {Q} ({\sqrt {D}}\,)$ has at most six units. In the case of the Gaussian integers (D = −1), the four units are 1, −1, √−1, −√−1. In the case of the Eisenstein integers (D = −3), the six units are ±1, ±1 ± √−3/2. For all other negative D, there are only two units, which are 1 and −1.
If D > 0, the ring of the integers of $\mathbb {Q} ({\sqrt {D}}\,)$ has infinitely many units that are equal to ± ui, where i is an arbitrary integer, and u is a particular unit called a fundamental unit. Given a fundamental unit u, there are three other fundamental units, its conjugate ${\overline {u}},$ and also $-u$ and $-{\overline {u}}.$ Commonly, one calls "the fundamental unit" the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written as a + b√D, with a and b positive (integers or halves of integers).
The fundamental units for the 10 smallest positive square-free D are 1 + √2, 2 + √3, 1 + √5/2 (the golden ratio), 5 + 2√6, 8 + 3√7, 3 + √10, 10 + 3√11, 3 + √13/2, 15 + 4√14, 4 + √15. For larger D, the coefficients of the fundamental unit may be very large. For example, for D = 19, 31, 43, the fundamental units are respectively 170 + 39√19, 1520 + 273√31 and 3482 + 531√43.
Examples of complex quadratic integer rings
For D < 0, ω is a complex (imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex numbers.
• A classic example is $\mathbf {Z} [{\sqrt {-1}}\,]$, the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.[5]
• The elements in ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {-3}}\,)}=\mathbf {Z} \left[{{1+{\sqrt {-3}}} \over 2}\right]$ are called Eisenstein integers.
Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ 4) and Q(ζ 3) correspondingly. In contrast, Z[√−3] is not even a Dedekind domain.
Both above examples are principal ideal rings and also Euclidean domains for the norm. This is not the case for
${\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}}\,)}=\mathbf {Z} \left[{\sqrt {-5}}\,\right],$
which is not even a unique factorization domain. This can be shown as follows.
In ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}}\,)},$ we have
$9=3\cdot 3=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).$
The factors 3, $2+{\sqrt {-5}}$ and $2-{\sqrt {-5}}$ are irreducible, as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of ±1 being at least 4. Thus the factorization of 9 into irreducible factors is not unique.
The ideals $\langle 3,1+{\sqrt {-5}}\,\rangle $ and $\langle 3,1-{\sqrt {-5}}\,\rangle $ are not principal, as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.
Examples of real quadratic integer rings
For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell's equation X 2 − DY 2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4).
• For D = 5, ω = 1+√5/2 is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet. Its units have the form ±ωn, where n is an arbitrary integer. This ring also arises from studying 5-fold rotational symmetry on Euclidean plane, for example, Penrose tilings.[6]
• Indian mathematician Brahmagupta treated the Pell's equation X 2 − 61Y 2 = 1, corresponding to the ring is Z[√61]. Some results were presented to European community by Pierre Fermat in 1657.
Principal rings of quadratic integers
The unique factorization property is not always verified for rings of quadratic integers, as seen above for the case of Z[√−5]. However, as for every Dedekind domain, a ring of quadratic integers is a unique factorization domain if and only if it is a principal ideal domain. This occurs if and only if the class number of the corresponding quadratic field is one.
The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}$ for
D = −1, −2, −3, −7, −11, −19, −43, −67, −163.
This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967 (see Stark–Heegner theorem). This is a special case of the famous class number problem.
There are many known positive integers D > 0, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.
Euclidean rings of quadratic integers
See also: Euclidean domain § Norm-Euclidean fields
When a ring of quadratic integers is a principal ideal domain, it is interesting to know whether it is a Euclidean domain. This problem has been completely solved as follows.
Equipped with the norm $N(a+b{\sqrt {D}}\,)=|a^{2}-Db^{2}|$ as a Euclidean function, ${\mathcal {O}}_{\mathbf {Q} ({\sqrt {D}}\,)}$ is a Euclidean domain for negative D when
D = −1, −2, −3, −7, −11,[7]
and, for positive D, when
D = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in the OEIS).
There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.[8]
For negative D, a ring of quadratic integers is Euclidean if and only if the norm is a Euclidean function for it. It follows that, for
D = −19, −43, −67, −163,
the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains.
On the other hand, the generalized Riemann hypothesis implies that a ring of real quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function, which can indeed differ from the usual norm.[9] The values D = 14, 69 were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.[10][11]
Notes
1. Dedekind 1871, Supplement X, p. 447
2. Bourbaki 1994, p. 99
3. "Why is quadratic integer ring defined in that way?". math.stackexchange.com. Retrieved 2016-12-31.
4. M. Artin, Algebra (2nd ed) Ch 13
5. Dummit, pg. 229
6. de Bruijn, N. G. (1981), "Algebraic theory of Penrose's non-periodic tilings of the plane, I, II" (PDF), Indagationes Mathematicae, 43 (1): 39–66
7. Dummit, pg. 272
8. LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57, 81. ISBN 978-0-486-42539-9. Zbl 1009.11001.
9. P. Weinberger, On Euclidean rings of algebraic integers. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321–332.
10. M. Harper, $\mathbb {Z} [{\sqrt {14}}]$ is Euclidean. Can. J. Math. 56(2004), 55–70.
11. David A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta Mathematica, 83(1994), 327–330 Archived 2015-01-29 at the Wayback Machine
References
• Bourbaki, Nicolas (1994). Elements of the history of mathematics. Translated by Meldrum, John. Berlin: Springer-Verlag. ISBN 978-3-540-64767-6. MR 1290116.
• Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg. Retrieved 5. August 2009
• Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed.
• Artin, M, Algebra, 2nd ed., Ch 13.
Further reading
• J.S. Milne. Algebraic Number Theory, Version 3.01, September 28, 2008. online lecture notes
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Wikipedia
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Quadratic integral
In mathematics, a quadratic integral is an integral of the form
$\int {\frac {dx}{a+bx+cx^{2}}}.$
It can be evaluated by completing the square in the denominator.
$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{\!2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.$
Positive-discriminant case
Assume that the discriminant q = b2 − 4ac is positive. In that case, define u and A by
$u=x+{\frac {b}{2c}},$
and
$-A^{2}={\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}={\frac {1}{4c^{2}}}(4ac-b^{2}).$
The quadratic integral can now be written as
$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {du}{u^{2}-A^{2}}}={\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}.$
The partial fraction decomposition
${\frac {1}{(u+A)(u-A)}}={\frac {1}{2A}}\!\left({\frac {1}{u-A}}-{\frac {1}{u+A}}\right)$
allows us to evaluate the integral:
${\frac {1}{c}}\int {\frac {du}{(u+A)(u-A)}}={\frac {1}{2Ac}}\ln \left({\frac {u-A}{u+A}}\right)+{\text{constant}}.$
The final result for the original integral, under the assumption that q > 0, is
$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{\sqrt {q}}}\ln \left({\frac {2cx+b-{\sqrt {q}}}{2cx+b+{\sqrt {q}}}}\right)+{\text{constant}}.$
Negative-discriminant case
In case the discriminant q = b2 − 4ac is negative, the second term in the denominator in
$\int {\frac {dx}{a+bx+cx^{2}}}={\frac {1}{c}}\int {\frac {dx}{\left(x+{\frac {b}{2c}}\right)^{\!2}+\left({\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}\right)}}.$
is positive. Then the integral becomes
${\begin{aligned}{\frac {1}{c}}\int {\frac {du}{u^{2}+A^{2}}}&={\frac {1}{cA}}\int {\frac {du/A}{(u/A)^{2}+1}}\\[9pt]&={\frac {1}{cA}}\int {\frac {dw}{w^{2}+1}}\\[9pt]&={\frac {1}{cA}}\arctan(w)+\mathrm {constant} \\[9pt]&={\frac {1}{cA}}\arctan \left({\frac {u}{A}}\right)+{\text{constant}}\\[9pt]&={\frac {1}{c{\sqrt {{\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}}}}}\arctan \left({\frac {x+{\frac {b}{2c}}}{\sqrt {{\frac {a}{c}}-{\frac {b^{2}}{4c^{2}}}}}}\right)+{\text{constant}}\\[9pt]&={\frac {2}{\sqrt {4ac-b^{2}\,}}}\arctan \left({\frac {2cx+b}{\sqrt {4ac-b^{2}}}}\right)+{\text{constant}}.\end{aligned}}$
References
• Weisstein, Eric W. "Quadratic Integral." From MathWorld--A Wolfram Web Resource, wherein the following is referenced:
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.
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Quadratic integrate and fire
The quadratic integrate and fire (QIF) model is a biological neuron model and a type of integrate-and-fire neuron which describes action potentials in neurons. In contrast to physiologically accurate but computationally expensive neuron models like the Hodgkin–Huxley model, the QIF model seeks only to produce action potential-like patterns and ignores subtleties like gating variables, which play an important role in generating action potentials in a real neuron. However, the QIF model is incredibly easy to implement and compute, and relatively straightforward to study and understand, thus has found ubiquitous use in computational neuroscience.[1]
A quadratic integrate and fire neuron is defined by the autonomous differential equation,
${\frac {dx}{dt}}=x^{2}+I$
where $I$ is a real positive constant. Note that a solution to this differential equation is the tangent function, which blows up in finite time. Thus a "spike" is said to have occurred when the solution reaches positive infinity, and the solution is reset to negative infinity.
When implementing this model in computers, a threshold crossing value ($V_{t}$) and a reset value ($V_{r}$) is assigned, so that when the solution rises above the threshold, $x(t)\geq V_{t}$, the solution is immediately reset to $V_{r}$
References
1. Fourcaud-Trocmé, Nicolas (2013), "Integrate and Fire Models, Deterministic", in Jaeger, Dieter; Jung, Ranu (eds.), Encyclopedia of Computational Neuroscience, New York, NY: Springer, pp. 1–9, doi:10.1007/978-1-4614-7320-6_148-1, ISBN 978-1-4614-7320-6, retrieved 2023-03-01
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Wikipedia
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Quadratic irrational number
In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers.[1] Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as
${a+b{\sqrt {c}} \over d},$
for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers. This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set.
Quadratic irrationals are used in field theory to construct field extensions of the field of rational numbers Q. Given the square-free integer c, the augmentation of Q by quadratic irrationals using √c produces a quadratic field Q(√c). For example, the inverses of elements of Q(√c) are of the same form as the above algebraic numbers:
${d \over a+b{\sqrt {c}}}={ad-bd{\sqrt {c}} \over a^{2}-b^{2}c}.$
Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all real quadratic irrationals, and only real quadratic irrationals, have periodic continued fraction forms. For example
${\sqrt {3}}=1.732\ldots =[1;1,2,1,2,1,2,\ldots ]$
The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers. The correspondence is explicitly provided by Minkowski's question mark function, and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map $h(x)=1/x-\lfloor 1/x\rfloor $ for continued fractions.
Real quadratic irrational numbers and indefinite binary quadratic forms
We may rewrite a quadratic irrationality as follows:
${\frac {a+b{\sqrt {c}}}{d}}={\frac {a+{\sqrt {b^{2}c}}}{d}}.$
It follows that every quadratic irrational number can be written in the form
${\frac {a+{\sqrt {c}}}{d}}.$
This expression is not unique.
Fix a non-square, positive integer $c$ congruent to $0$ or $1$ modulo $4$, and define a set $S_{c}$ as
$S_{c}=\left\{{\frac {a+{\sqrt {c}}}{d}}\colon a,d{\text{ integers, }}\,d{\text{ even}},\,a^{2}\equiv c{\pmod {2d}}\right\}.$
Every quadratic irrationality is in some set $S_{c}$, since the congruence conditions can be met by scaling the numerator and denominator by an appropriate factor.
A matrix
${\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}$
with integer entries and $\alpha \delta -\beta \gamma =1$ can be used to transform a number $y$ in $S_{c}$. The transformed number is
$z={\frac {\alpha y+\beta }{\gamma y+\delta }}$
If $y$ is in $S_{c}$, then $z$ is too.
The relation between $y$ and $z$ above is an equivalence relation. (This follows, for instance, because the above transformation gives a group action of the group of integer matrices with determinant 1 on the set $S_{c}$.) Thus, $S_{c}$ partitions into equivalence classes. Each equivalence class comprises a collection of quadratic irrationalities with each pair equivalent through the action of some matrix. Serret's theorem implies that the regular continued fraction expansions of equivalent quadratic irrationalities are eventually the same, that is, their sequences of partial quotients have the same tail. Thus, all numbers in an equivalence class have continued fraction expansions that are eventually periodic with the same tail.
There are finitely many equivalence classes of quadratic irrationalities in $S_{c}$. The standard proof of this involves considering the map $\phi $ from binary quadratic forms of discriminant $c$ to $S_{c}$ given by
$\phi (tx^{2}+uxy+vy^{2})={\frac {-u+{\sqrt {c}}}{2t}}$
A computation shows that $\phi $ is a bijection that respects the matrix action on each set. The equivalence classes of quadratic irrationalities are then in bijection with the equivalence classes of binary quadratic forms, and Lagrange showed that there are finitely many equivalence classes of binary quadratic forms of given discriminant.
Through the bijection $\phi $, expanding a number in $S_{c}$ in a continued fraction corresponds to reducing the quadratic form. The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.
Square root of non-square is irrational
The definition of quadratic irrationals requires them to satisfy two conditions: they must satisfy a quadratic equation and they must be irrational. The solutions to the quadratic equation ax2 + bx + c = 0 are
${\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.$
Thus quadratic irrationals are precisely those real numbers in this form that are not rational. Since b and 2a are both integers, asking when the above quantity is irrational is the same as asking when the square root of an integer is irrational. The answer to this is that the square root of any natural number that is not a square number is irrational.
The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to classification of irrational magnitudes. The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma.
Many proofs of the irrationality of the square roots of non-square natural numbers implicitly assume the fundamental theorem of arithmetic, which was first proven by Carl Friedrich Gauss in his Disquisitiones Arithmeticae. This asserts that every integer has a unique factorization into primes. For any rational non-integer in lowest terms there must be a prime in the denominator which does not divide into the numerator. When the numerator is squared that prime will still not divide into it because of the unique factorization. Therefore, the square of a rational non-integer is always a non-integer; by contrapositive, the square root of an integer is always either another integer, or irrational.
Euclid used a restricted version of the fundamental theorem and some careful argument to prove the theorem. His proof is in Euclid's Elements Book X Proposition 9.[2]
The fundamental theorem of arithmetic is not actually required to prove the result, however. There are self-contained proofs by Richard Dedekind,[3] among others. The following proof was adapted by Colin Richard Hughes from a proof of the irrationality of the square root of 2 found by Theodor Estermann in 1975.[4][5]
If D is a non-square natural number, then there is a number n such that:
n2 < D < (n + 1)2,
so in particular
0 < √D − n < 1.
If the square root of D is rational, then it can be written as the irreducible fraction p/q, so that q is the smallest possible denominator, and hence the smallest number for which q√D is also an integer. Then:
(√D − n)q√D = qD − nq√D
which is thus also an integer. But 0 < (√D − n) < 1 so (√D − n)q < q. Hence (√D − n)q is an integer smaller than q which multiplied by √D makes an integer. This is a contradiction, because q was defined to be the smallest such number. Therefore, √D cannot be rational.
See also
• Algebraic number field
• Apotome (mathematics)
• Periodic continued fraction
• Restricted partial quotients
• Quadratic integer
References
1. Jörn Steuding, Diophantine Analysis, (2005), Chapman & Hall, p.72.
2. Euclid. "Euclid's Elements Book X Proposition 9". D.E.Joyce, Clark University. Retrieved 2008-10-29.
3. Bogomolny, Alexander. "Square root of 2 is irrational". Interactive Mathematics Miscellany and Puzzles. Retrieved May 5, 2016.
4. Hughes, Colin Richard (1999). "Irrational roots". Mathematical Gazette. 83 (498): 502–503. doi:10.2307/3620972. JSTOR 3620972. S2CID 149602021.
5. Estermann, Theodor (1975). "The irrationality of √2". Mathematical Gazette. 59 (408): 110. doi:10.2307/3616647. JSTOR 3616647. S2CID 126072097.
External links
• Weisstein, Eric W. "Quadratic irrational number". MathWorld.
• Continued fraction calculator for quadratic irrationals
• Proof that e is not a quadratic irrational
Algebraic numbers
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Wikipedia
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Quadratic knapsack problem
The quadratic knapsack problem (QKP), first introduced in 19th century,[1] is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as to maximize the overall profit. Usually, quadratic knapsack problems come with a restriction on the number of copies of each kind of item: either 0, or 1. This special type of QKP forms the 0-1 quadratic knapsack problem, which was first discussed by Gallo et al.[2] The 0-1 quadratic knapsack problem is a variation of knapsack problems, combining the features of unbounded knapsack problem, 0-1 knapsack problem and quadratic knapsack problem.
Definition
Specifically, the 0–1 quadratic knapsack problem has the following form:
${\text{maximize }}\left\{\sum _{i=1}^{n}p_{i}x_{i}+\sum _{i=1}^{n}\sum _{j=1,i\neq j}^{n}P_{ij}x_{i}x_{j}:x\in X,x{\text{ binary}}\right\}$
${\text{subject to }}X\equiv \left\{x\in \{0,1\}^{n}:\sum _{i=1}^{n}w_{i}x_{i}\leq W;x_{i}\in \{0,1\}{\text{ for }}i=1,\ldots ,n\right\}.$
Here the binary variable xi represents whether item i is included in the knapsack, $p_{i}$ is the profit earned by selecting item i and $P_{ij}$ is the profit achieved if both item i and j are added.
Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity.
Application
As one might expect, QKP has a wide range of applications including telecommunication, transportation network, computer science and economics. In fact, Witzgall first discussed QKP when selecting sites for satellite stations in order to maximize the global traffic with respect to a budget constraint. Similar model applies to problems like considering the location of airports, railway stations, or freight handling terminals.[3] Applications of QKP in the field of computer science is more common after the early days: compiler design problem,[4] clique problem,[5][6] very large scale integration (VLSI) design.[7] Additionally, pricing problems appear to be an application of QKP as described by Johnson et al.[8]
Computational complexity
In general, the decision version of the knapsack problem (Can a value of at least V be achieved under a restriction of a certain capacity W?) is NP-complete.[9] Thus, a given solution can be verified to in polynomial time while no algorithm can identify a solution efficiently.
The optimization knapsack problem is NP-hard and there is no known algorithm that can solve the problem in polynomial time.
As a particular variation of the knapsack problem, the 0-1 quadratic knapsack problem is also NP-hard.
While no available efficient algorithm exists in the literature, there is a pseudo-polynomial time based on dynamic programming and other heuristic algorithms that can always generate “good” solutions.
Solving
While the knapsack problem is one of the most commonly solved operation research (OR) problems, there are limited efficient algorithms that can solve 0-1 quadratic knapsack problems. Available algorithms include but are not limited to brute force, linearization,[10] and convex reformulation. Just like other NP-hard problems, it is usually enough to find a workable solution even if it is not necessarily optimal. Heuristic algorithms based on greedy algorithm, dynamic programming can give a relatively “good” solution to the 0-1 QKP efficiently.
Brute force
The brute-force algorithm to solve this problem is to identify all possible subsets of the items without exceeding the capacity and select the one with the optimal value. The pseudo-code is provided as follows:
// Input:
// Profits (stored in array p)
// Quadratic profits (stored in matrix P)
// Weights (stored in array w)
// Number of items (n)
// Knapsack capacity (W)
int max = 0
for all subset S do
int value, weight = 0
for i from 0 to S.size-1 do:
value = value + p[i]
weight = weight + w[i]
for j from i+1 to S.size-1 do:
value = value + P[i][j]
if weight <= W then:
if value > max then:
max = value
Given n items, there will be at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): 2^{n} subsets and for each legal candidate set, the running time of computing the values earned is $O(n^{2})$. Thus, the efficiency class of brute-force algorithm is $(2^{n}n^{2})=\lambda (2^{n})$, being exponential.
Linearization
Problems of such form are difficult to solve directly using standard solvers and thus people try to reformulate it as a linear program using auxiliary variables and constraints so that the problem can be readily solved using commercial packages. Two well-known linearization approaches for the 0-1 QKP are the standard linearization and Glover’s linearization.[11][12][13]
Standard linearization
The first one is the standard linearization strategy, as shown below:
LP1: maximize
$\sum _{i=1}^{n}p_{i}x_{i}+\sum _{i=1}^{n}\left(\sum _{j=1,i<j}^{n}(P_{ij}+P_{ji})z_{ij}\right).$
subject to
$z_{ij}\leq x_{i}$ for all $(i,j),i<j$
$z_{ij}\leq x_{j}$ for all $(i,j),i<j$
$x_{i}+x_{j}-1\leq z_{ij}$ for all $(i,j),i<j$
$z_{ij}\geq 0$ for all $(i,j),i<j$
$x\in X,x$ binary
In the formulation LP1, we have replaced the xixj term with a continuous variable zij. This reformulates the QKP into a knapsack problem, which we can then solve optimally using standard solvers.
Glover's linearization
The second reformulation, which is more concise, is called Glover’s linearization.[14][15][16] The Glover formulation is shown below, where Li and Ui are lower and upper bounds on $\sum _{j=1,i\neq j}^{n}P_{ij}x_{j}$, respectively:
LP2: maximize
$\sum _{i=1}^{n}p_{i}x_{i}+\sum _{i=1}^{n}z_{i}$
subject to
$L_{i}x_{i}\leq z_{i}\leq U_{i}x_{i}$ for $i=1,\ldots ,n$
$\sum _{j=1,i\neq j}^{n}P_{ij}x_{j}-U_{i}(1-x_{i})\leq z_{i}\leq \sum _{j=1,i\neq j}^{n}P_{ij}x_{j}-L_{i}(1-x_{i})$ for $i=1,\ldots ,n$
$x\in X,x$ binary
In the formulation LP2, we have replaced the expression $\sum _{j=1,i\neq j}^{n}P_{ij}x_{i}x_{j}$ with a continuous variable zi. Similarly, we can use standard solvers to solve the linearized problem. Note that Glover’s linearization only includes $n$ auxiliary variables with $2n$ constraints while standard linearization requires ${n \choose 2}$ auxiliary variables and $3{n \choose 2}$ constraints to achieve linearity.
Convex quadratic reformulation
Note that nonlinear programs are hard to solve due to the possibility of being stuck at a local maximum. However, when the program is convex, any local maximum is the global maximum. A convex program is to maximize a concave function or minimize a convex function on a convex set. A set S is convex if $\forall u,v\in S$, $\lambda u+(1-\lambda )v\in S$ where $\lambda \in [0,1]$. That is to say, any point between two points in the set must also be an element of the set. A function f is concave if $f[\lambda u+(1-\lambda )v]\leq \lambda f(u)+(1-\lambda )f(v)$. A function f is convex if $f[\lambda u+(1-\lambda )v]\geq \lambda f(u)+(1-\lambda )f(v)$. Informally, a function is concave if the line segment connecting two points on the graph lies above or on the graph, while a function is convex if below or on the graph. Thus, by rewriting the objective function into an equivalent convex function, we can reformulate the program to be convex, which can be solved using optimization packages.
The objective function can be written as $c^{T}x+x^{T}Cx$ using linear algebra notation. We need to make P a positive semi-definite matrix in order to reformulate a convex function. In this case, we modify the objective function to be $p^{T}x+x^{T}Px+\sum _{i=1}^{n}\left(\sum _{j=1,j\neq i}^{n}|P_{ij}|\right)(x_{i}^{2}-x_{i})$ by applying results from linear algebra, where P is a diagonally dominant matrix and thus a positive semi-definite. This reformulation can be solved using a standard commercial mixed-integer quadratic package.[17]
Greedy heuristic algorithm
George Dantzig[18] proposed a greedy approximation algorithm to unbounded knapsack problem which can also be used to solve the 0-1 QKP. The algorithm consists of two phrases: identify an initial solution and improve it.
First compute for each item, the total objective contribution realizable by selecting it, $p_{i}+\sum _{i\neq j}^{n}P_{ij}$, and sort the items in decreasing order of the potential value per unit of weight, $(p_{i}+\sum _{i\neq j}^{n}P_{ij})/w_{i}$. Then select the items with the maximal value-weight ratio into the knapsack until there is no space for more, which forms the initial solution. Starting with the initial solution, the improvement is conducted by pairwise exchange. For each item in the solution set, identify the items not in the set where swapping results in an improving objective. Select the pair with maximal improvement and swap. There are also possibilities that removing one from the set or adding one to the set will produce the greatest contribution. Repeat until there is no improving swapping. The complexity class of this algorithm is $O(2^{n})$ since for the worst case every possible combination of items will be identified.
Quadknap
Quadknap is an exact branch-and-bound algorithm proposed by Caprara et al.,[19] where upper bounds are computed by considering a Lagrangian relaxation which approximate a difficult problem by a simpler problem and penalizes violations of constraints using Lagrange multiplier to impost a cost on violations. Quadknap releases the integer requirement when computing the upper bounds. Suboptimal Lagrangian multipliers are derived from sub-gradient optimization and provide a convenient reformulation of the problem. This algorithm is quite efficient since Lagrangian multipliers are stable, and suitable data structures are adopted to compute a tight upper bound in linear expected time in the number of variables. This algorithm was reported to generate exact solutions of instances with up to 400 binary variables, i.e., significantly larger than those solvable by other approaches. The code was written in C and is available online.[20]
Dynamic programming heuristic
While dynamic programming can generate optimal solutions to knapsack problems, dynamic programming approaches for QKP[21] can only yield a relatively good quality solution, which can serve as a lower bound to the optimal objectives. While it runs in pseudo-polynomial time, it has a large memory requirement.
Dynamic programming algorithm
For simplicity, assume all weights are non-negative. The objective is to maximize total value subject to the constraint: that the total weight is less than or equal to W. Then for each $w\leq W$, define $f(m,w)$ to be the value of the most profitable packing of the first m items found with a total weight of w. That is, let
$f(m,w)=\max \left\{\sum _{i=1}^{m}p_{i}x_{i}+\sum _{i=1}^{m}\sum _{j=1,i\neq j}^{m}P_{ij}x_{i}x_{j}:\sum _{i=1}^{m}w_{i}=w,1\leq i\leq m\right\}.$
Then, $f(m,w)$ is the solution to the problem. Note that by dynamic programming, the solution to a problem arises from the solution to its smaller sub-problems. In this particular case, start with the first item and try to find a better packing by considering adding items with an expected weight of 𝑤. If the weight of the item to be added exceeds 𝑤, then $f(m,w)$ is the same with $f(m-1,w)$. Given that the item has a smaller weight compared with the desired weight, $f(m,w)$ is either the same as $f(m-1,w)$ if adding makes no contribution, or the same as the solution for a knapsack with smaller capacity, specifically one with the capacity reduced by the weight of that chosen item, plus the value of one correct item, i.e. $f(m-1,w-w_{m})+p_{m}+\sum _{i=1}^{m-1}P_{im}x_{i}$. To conclude, we have that
$f(m,w)={\begin{cases}\max f(m-1,w),f(m-1,w-w_{m})+p_{m}+\sum _{i=1}^{m-1}P_{im}x_{i}&{\text{if }}w_{m}\leq w\\f(m-1,w)&{\text{otherwise}}\end{cases}}$
Note on efficiency class: Clearly the running time of this algorithm is $O(Wn^{2})$, based on the nested loop and the computation of the profit of new packing. This does not contradict the fact the QKP is NP-hard since W is not polynomial in the length of the input.
Revised dynamic programming algorithm
Note that the previous algorithm requires $O(Wn^{2})$ space for storing the current packing of items for all m,w, which may not be able to handle large-size problems. In fact, this can be easily improved by dropping the index m from $f(m,w)$ since all the computations depend only on the results from the preceding stage.
Redefine $f(w)$ to be the current value of the most profitable packing found by the heuristic. That is,
$f(w)=\max \left\{\sum _{i=1}^{m}p_{i}x_{i}+\sum _{i=1}^{m}\sum _{j=1,i\neq j}^{m}P_{ij}x_{i}x_{j}:\sum _{i=1}^{m}w_{i}=w,m\leq n\right\}.$
Accordingly, by dynamic programming we have that
$f(m)={\begin{cases}\max f(w),f(w-w_{m})+p_{m}+\sum _{i=1}^{m-1}P_{im}x_{i}&{\text{if }}w_{m}\leq w,\\f(w)&{\text{otherwise.}}\end{cases}}$
Note this revised algorithm still runs in $O(Wn^{2})$ while only taking up $O(Wn)$ memory compared to the previous $O(Wn^{2})$.
Related research topics
Researchers have studied 0-1 quadratic knapsack problems for decades. One focus is to find effective algorithms or effective heuristics, especially those with an outstanding performance solving real world problems. The relationship between the decision version and the optimization version of the 0-1 QKP should not be ignored when working with either one. On one hand, if the decision problem can be solved in polynomial time, then one can find the optimal solution by applying this algorithm iteratively. On the other hand, if there exists an algorithm that can solve the optimization problem efficiently, then it can be utilized in solving the decision problem by comparing the input with the optimal value.
Another theme in literature is to identify what are the "hard" problems. Researchers who study the 0-1 QKP often perform computational studies[22] to show the superiority of their strategies. Such studies can also be conducted to assess the performance of different solution methods. For the 0-1 QKP, those computational studies often rely on randomly generated data, introduced by Gallo et al. Essentially every computational study of the 0-1 QKP utilizes data that is randomly generated as follows. The weights are integers taken from a uniform distribution over the interval [1, 50], and the capacity constraints is an integer taken from a uniform distribution between 50 and the sum of item weights. The objective coefficients, i.e. the values are randomly chosen from [1,100]. It has been observed that generating instances of this form yields problems with highly variable and unpredictable difficulty. Therefore, the computational studies presented in the literature may be unsound. Thus some researches aim to develop a methodology to generate instances of the 0-1 QKP with a predictable and consistent level of difficulty.
See also
• Knapsack problem
• Combinatorial auction
• Combinatorial optimization
• Continuous knapsack problem
• List of knapsack problems
• Packing problem
Notes
1. C., Witzgall (1975). "Mathematical methods of site selection for Electronic Message Systems (EMS)". NBS Internal Report. 76: 18321. Bibcode:1975STIN...7618321W. doi:10.6028/nbs.ir.75-737.
2. Gallo, G.; Hammer, P.L.; Simeone, B. (1980). Quadratic knapsack problems. pp. 132–149. doi:10.1007/bfb0120892. ISBN 978-3-642-00801-6. {{cite book}}: |journal= ignored (help)
3. Rhys, J.M.W. (1970). "A Selection Problem of Shared Fixed Costs and Network Flows". Management Science. 17 (3): 200–207. doi:10.1287/mnsc.17.3.200.
4. Helmberg, C.; Rendl, F.; Weismantel, R. (1996). Quadratic knapsack relaxations using cutting planes and semidefinite programming. pp. 175–189. doi:10.1007/3-540-61310-2_14. ISBN 978-3-540-61310-7. {{cite book}}: |journal= ignored (help)
5. Dijkhuizen, G.; Faigle, U. (1993). "A cutting-plane approach to the edge-weighted maximal clique problem". European Journal of Operational Research. 69 (1): 121–130. doi:10.1016/0377-2217(93)90097-7.
6. Park, Kyungchul; Lee, Kyungsik; Park, Sungsoo (1996). "An extended formulation approach to the edge-weighted maximal clique problem". European Journal of Operational Research. 95 (3): 671–682. doi:10.1016/0377-2217(95)00299-5.
7. Ferreira, C.E.; Martin, A.; Souza, C.C.De; Weismantel, R.; Wolsey, L.A. (1996). "Formulations and valid inequalities for the node capacitated graph partitioning problem". Mathematical Programming. 74 (3): 247–266. doi:10.1007/bf02592198. S2CID 37819561.
8. Johnson, Ellis L.; Mehrotra, Anuj; Nemhauser, George L. (1993). "Min-cut clustering". Mathematical Programming. 62 (1–3): 133–151. doi:10.1007/bf01585164. S2CID 39694326.
9. Garey, Michael R.; Johnson, David S. (1979). Computers and intractibility: A guide to the theory of NP completeness. New York: Freeman and Co.
10. Adams, Warren P.; Sherali, Hanif D. (1986). "A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems". Management Science. 32 (10): 1274–1290. doi:10.1287/mnsc.32.10.1274.
11. Adams, Warren P.; Forrester, Richard J.; Glover, Fred W. (2004). "Comparisons and enhancement strategies for linearizing mixed 0-1 quadratic programs". Discrete Optimization. 1 (2): 99–120. doi:10.1016/j.disopt.2004.03.006.
12. Adams, Warren P.; Forrester, Richard J. (2005). "A simple recipe for concise mixed 0-1 linearizations". Operations Research Letters. 33 (1): 55–61. doi:10.1016/j.orl.2004.05.001.
13. Adams, Warren P.; Forrester, Richard J. (2007). "Linear forms of nonlinear expressions: New insights on old ideas". Operations Research Letters. 35 (4): 510–518. doi:10.1016/j.orl.2006.08.008.
14. Glover, Fred; Woolsey, Eugene (1974). "Technical Note—Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program". Operations Research. 22 (1): 180–182. doi:10.1287/opre.22.1.180.
15. Glover, Fred (1975). "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems". Management Science. 22 (4): 455–460. doi:10.1287/mnsc.22.4.455. S2CID 17004334.
16. Glover, Fred; Woolsey, Eugene (1973). "Further Reduction of Zero-One Polynomial Programming Problems to Zero-One linear Programming Problems". Operations Research. 21 (1): 156–161. doi:10.1287/opre.21.1.156.
17. Bliek, Christian; Bonami, Pierre; Lodi, Andrea (2014). "Solving Mixed-Integer Quadratic Programming problems with IBM-CPLEX: a progress report" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
18. Dantzig, George B. (1957). "Discrete-Variable Extremum Problems". Operations Research. 5 (2): 266–288. doi:10.1016/j.disopt.2004.03.006.
19. Caprara, Alberto; Pisinger, David; Toth, Paolo (1999). "Exact Solution of the Quadratic Knapsack Problem". INFORMS Journal on Computing. 11 (2): 125–137. CiteSeerX 10.1.1.22.2818. doi:10.1287/ijoc.11.2.125.
20. "Quadknap". Retrieved 2016-12-03.
21. Fomeni, Franklin Djeumou; Letchford, Adam N. (2014). "A Dynamic Programming Heuristic for the Quadratic Knapsack Problem". INFORMS Journal on Computing. 26 (1): 173–182. doi:10.1287/ijoc.2013.0555. S2CID 15570245.
22. Forrester, Richard J.; Adams, Warren P.; Hadavas, Paul T. (2009). "Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem". Naval Research Logistics. 57: 1–12. doi:10.1002/nav.20364. S2CID 121015443.
External links
• David Pisinger's Codes for different knapsack problem
• Codes for Quadratic Knapsack Problem
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Wikipedia
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Quadratic pair
In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by Thompson (1971), is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimal polynomial (x − 1)2. Thompson classified the quadratic pairs for p ≥ 5. Chermak (2004) classified the quadratic pairs for p = 3. With a few exceptions, especially for p = 3, groups with a quadratic pair for the prime p tend to be more or less groups of Lie type in characteristic p.
See also
p-stable group
References
• Chermak, Andrew (2004), "Quadratic pairs", Journal of Algebra, 277 (1): 36–72, doi:10.1016/S0021-8693(03)00334-X, ISSN 0021-8693, MR 2059620
• Thompson, John G. (1971), "Quadratic pairs", Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 1, Gauthier-Villars, pp. 375–376, MR 0430043
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Wikipedia
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Quadratic pseudo-Boolean optimization
Quadratic pseudo-Boolean optimisation (QPBO) is a combinatorial optimization method for quadratic pseudo-Boolean functions in the form
$f(\mathbf {x} )=w_{0}+\sum _{p\in V}w_{p}(x_{p})+\sum _{(p,q)\in E}w_{pq}(x_{p},x_{q})$
in the binary variables $x_{p}\in \{0,1\}\;\forall p\in V=\{1,\dots ,n\}$, with $E\subseteq V\times V$. If $f$ is submodular then QPBO produces a global optimum equivalently to graph cut optimization, while if $f$ contains non-submodular terms then the algorithm produces a partial solution with specific optimality properties, in both cases in polynomial time.[1]
QPBO is a useful tool for inference on Markov random fields and conditional random fields, and has applications in computer vision problems such as image segmentation and stereo matching.[2]
Optimization of non-submodular functions
If the coefficients $w_{pq}$ of the quadratic terms satisfy the submodularity condition
$w_{pq}(0,0)+w_{pq}(1,1)\leq w_{pq}(0,1)+w_{pq}(1,0)$
then the function can be efficiently optimised with graph cut optimization. It is indeed possible to represent it with a non-negative weighted graph, and the global minimum can be found in polynomial time by computing a minimum cut of the graph, which can be computed with algorithms such as Ford–Fulkerson, Edmonds–Karp, and Boykov–Kolmogorov's.
If the function is not submodular, then the problem is NP-hard in the general case and it is not always possible to solve it exactly in polynomial time. It is possible to replace the target function with a similar but submodular approximation, e.g. by removing all non-submodular terms or replacing them with submodular approximations, but such approach is generally sub-optimal and it produces satisfying results only if the number of non-submodular terms is relatively small.[1]
QPBO builds an extended graph, introducing a set of auxiliary variables ideally equivalent to the negation of the variables in the problem. If the nodes in the graph associated to a variable (representing the variable itself and its negation) are separated by the minimum cut of the graph in two different connected components, then the optimal value for such variable is well defined, otherwise it is not possible to infer it. Such method produces results generally superior to submodular approximations of the target function.[1]
Properties
QPBO produces a solution where each variable assumes one of three possible values: true, false, and undefined, noted in the following as 1, 0, and $\emptyset $ respectively. The solution has the following two properties.
• Partial optimality: if $f$ is submodular, then QPBO produces a global minimum exactly, equivalent to graph cut, and all variables have a non-undefined value; if submodularity is not satisfied, the result will be a partial solution $\mathbf {x} $ where a subset ${\hat {V}}\subseteq V$ of the variables have a non-undefined value. A partial solution is always part of a global solution, i.e. there exists a global minimum point $\mathbf {x^{*}} $ for $f$ such that $x_{i}=x_{i}^{*}$ for each $i\in {\hat {V}}$.
• Persistence: given a solution $\mathbf {x} $ generated by QPBO and an arbitrary assignment of values $\mathbf {y} $ to the variables, if a new solution ${\hat {\mathbf {y} }}$ is constructed by replacing $y_{i}$ with $x_{i}$ for each $i\in {\hat {V}}$, then $f({\hat {\mathbf {y} }})\leq f(\mathbf {y} )$.[1]
Algorithm
The algorithm can be divided in three steps: graph construction, max-flow computation, and assignment of values to the variables.
When constructing the graph, the set of vertices $V$ contains the source and sink nodes $s$ and $t$, and a pair of nodes $p$ and $p'$ for each variable. After re-parametrising the function to normal form,[note 1] a pair of edges is added to the graph for each term $w$:
• for each term $w_{p}(0)$ the edges $p\rightarrow t$ and $s\rightarrow p'$, with weight ${\frac {1}{2}}w_{p}(0)$;
• for each term $w_{p}(1)$ the edges $s\rightarrow p$ and $p'\rightarrow t$, with weight ${\frac {1}{2}}w_{p}(1)$;
• for each term $w_{pq}(0,1)$ the edges $p\rightarrow q$ and $q'\rightarrow p'$, with weight ${\frac {1}{2}}w_{pq}(0,1)$;
• for each term $w_{pq}(1,0)$ the edges $q\rightarrow p$ and $p'\rightarrow q'$, with weight ${\frac {1}{2}}w_{pq}(1,0)$;
• for each term $w_{pq}(0,0)$ the edges $p\rightarrow q'$ and $q\rightarrow p'$, with weight ${\frac {1}{2}}w_{pq}(0,0)$;
• for each term $w_{pq}(1,1)$ the edges $q'\rightarrow p$ and $p'\rightarrow q$, with weight ${\frac {1}{2}}w_{pq}(1,1)$.
The minimum cut of the graph can be computed with a max-flow algorithm. In the general case, the minimum cut is not unique, and each minimum cut correspond to a different partial solution, however it is possible to build a minimum cut such that the number of undefined variables is minimal.
Once the minimum cut is known, each variable receives a value depending upon the position of its corresponding nodes $p$ and $p'$: if $p$ belongs to the connected component containing the source and $p'$ belongs to the connected component containing the sink then the variable will have value of 0. Vice versa, if $p$ belongs to the connected component containing the sink and $p'$ to the one containing the source, then the variable will have value of 1. If both nodes $p$ and $p'$ belong to the same connected component, then the value of the variable will be undefined.[2]
The way undefined variables can be handled is dependent upon the context of the problem. In the general case, given a partition of the graph in two sub-graphs and two solutions, each one optimal for one of the sub-graphs, then it is possible to combine the two solutions into one solution optimal for the whole graph in polynomial time.[3] However, computing an optimal solution for the subset of undefined variables is still a NP-hard problem. In the context of iterative algorithms such as $\alpha $-expansion, a reasonable approach is to leave the value of undefined variables unchanged, since the persistence property guarantees that the target function will have non-increasing value.[1] Different exact and approximate strategies to minimise the number of undefined variables exist.[2]
Higher order terms
It is always possible to reduce a higher-order function to a quadratic function which is equivalent with respect to the optimisation, problem known as "higher-order clique reduction" (HOCR), and the result of such reduction can be optimized with QPBO. Generic methods for reduction of arbitrary functions rely on specific substitution rules and in the general case they require the introduction of auxiliary variables.[4] In practice most terms can be reduced without introducing additional variables, resulting in a simpler optimization problem, and the remaining terms can be reduced exactly, with addition of auxiliary variables, or approximately, without addition of any new variable.[5]
Notes
1. Kolmogorov and Rother (2007).
2. Rother et al. (2007).
3. Billionnet and Jaumard (1989).
4. Fix et al. (2011).
5. Ishikawa (2014).
References
• Billionnet, Alain; Jaumard, Brigitte (1989). "A decomposition method for minimizing quadratic pseudo-boolean functions". Operations Research Letters. 8 (3): 161–163. doi:10.1016/0167-6377(89)90043-6.
• Fix, Alexander; Gruber, Aritanan; Boros, Endre; Zabih, Ramin (2011). A graph cut algorithm for higher-order Markov random fields (PDF). International Conference on Computer Vision. pp. 1020–1027.
• Ishikawa, Hiroshi (2014). Higher-Order Clique Reduction Without Auxiliary Variables (PDF). Conference on Computer Vision and Pattern Recognition. IEEE. pp. 1362–1269.
• Kolmogorov, Vladimir; Rother, Carsten (2007). "Minimizing Nonsubmodular Functions: A Review". IEEE Transactions on Pattern Analysis and Machine Intelligence. IEEE. 29 (7): 1274–1279. doi:10.1109/tpami.2007.1031. PMID 17496384.
• Rother, Carsten; Kolmogorov, Vladimir; Lempitsky, Victor; Szummer, Martin (2007). Optimizing binary MRFs via extended roof duality (PDF). Conference on Computer Vision and Pattern Recognition. pp. 1–8.
Notes
1. The representation of a pseudo-Boolean function with coefficients $\mathbf {w} =(w_{0},w_{1},\dots ,w_{nn})$ is not unique, and if two coefficient vectors $\mathbf {w} $ and $\mathbf {w} '$ represent the same function then $\mathbf {w} '$ is said to be a reparametrisation of $\mathbf {w} $ and vice versa. In some constructions it is useful to ensure that the function has a specific form, called normal form, which is always defined for any function, and it is not unique. A function $f$ is in normal form if the two following conditions hold (Kolmogorov and Rother (2007)):
1. $\min\{w_{p}^{0},w_{p}^{1}\}=0$ for each $p\in V$;
2. $\min\{w_{pq}^{0j},w_{pq}^{1j}\}=0$ for each $(p,q)\in E$ and for each $j\in \{0,1\}$.
Given an arbitrary function $f$, it is always possible to find a reparametrisation to normal form with the following algorithm in two steps (Kolmogorov and Rother (2007)):
1. as long as there exist indices $(p,q)\in E$ and $j\in \{0,1\}$ such that the second condition of normality is not satisfied, substitute:
• $w_{pq}^{0j}$ with $w_{pq}^{0j}-a$
• $w_{pq}^{1j}$ with $w_{pq}^{1j}-a$
• $w_{q}^{j}$ with $w_{q}^{j}+a$
where $a=\min\{w_{pq}^{0j},w_{pq}^{1j}\}$;
2. for $p=1,\dots ,n$, substitute:
• $w_{0}$ with $w_{0}+a$
• $w_{p}^{0}$ with $w_{p}^{0}-a$
• $w_{p}^{1}$ with $w_{p}^{1}-a$
where $a=\min\{w_{p}^{0},w_{p}^{1}\}$.
External links
• Implementation of QPBO (C++), available under the GNU General Public License, by Vladimir Kolmogorov.
• Implementation of HOCR (C++), available under the MIT license, by Hiroshi Ishikawa.
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Wikipedia
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Quadratic quadrilateral element
The quadratic quadrilateral element, also known as the Q8 element is a type of element used in finite element analysis which is used to approximate in a 2D domain the exact solution to a given differential equation. It is a two-dimensional finite element with both local and global coordinates. This element can be used for plane stress or plane strain problems in elasticity. The quadratic quadrilateral element has modulus of elasticity E, Poisson’s ratio v, and thickness t.[1]
References
1. Kattan, Peter I. (2008), "The Quadratic Quadrilateral Element", MATLAB Guide to Finite Elements, Springer Berlin Heidelberg, pp. 311–336, doi:10.1007/978-3-540-70698-4_14, ISBN 9783540706977
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Wikipedia
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Quadratic residuosity problem
The quadratic residuosity problem (QRP[1]) in computational number theory is to decide, given integers $a$ and $N$, whether $a$ is a quadratic residue modulo $N$ or not. Here $N=p_{1}p_{2}$ for two unknown primes $p_{1}$ and $p_{2}$, and $a$ is among the numbers which are not obviously quadratic non-residues (see below).
The problem was first described by Gauss in his Disquisitiones Arithmeticae in 1801. This problem is believed to be computationally difficult. Several cryptographic methods rely on its hardness, see § Applications.
An efficient algorithm for the quadratic residuosity problem immediately implies efficient algorithms for other number theoretic problems, such as deciding whether a composite $N$ of unknown factorization is the product of 2 or 3 primes.[2]
Precise formulation
Given integers $a$ and $T$, $a$ is said to be a quadratic residue modulo $T$ if there exists an integer $b$ such that
$a\equiv b^{2}{\pmod {T}}$.
Otherwise we say it is a quadratic non-residue. When $T=p$ is a prime, it is customary to use the Legendre symbol:
$\left({\frac {a}{p}}\right)={\begin{cases}1&{\text{ if }}a{\text{ is a quadratic residue modulo }}p{\text{ and }}a\not \equiv 0{\pmod {p}},\\-1&{\text{ if }}a{\text{ is a quadratic non-residue modulo }}p,\\0&{\text{ if }}a\equiv 0{\pmod {p}}.\end{cases}}$
This is a multiplicative character which means ${\big (}{\tfrac {a}{p}}{\big )}=1$ for exactly $(p-1)/2$ of the values $1,\ldots ,p-1$, and it is $-1$ for the remaining.
It is easy to compute using the law of quadratic reciprocity in a manner akin to the Euclidean algorithm, see Legendre symbol.
Consider now some given $N=p_{1}p_{2}$ where $p_{1}$ and $p_{2}$ are two, different unknown primes. A given $a$ is a quadratic residue modulo $N$ if and only if $a$ is a quadratic residue modulo both $p_{1}$ and $p_{2}$ and $gcd(a,N)=1$.
Since we don't know $p_{1}$ or $p_{2}$, we cannot compute ${\big (}{\tfrac {a}{p_{1}}}{\big )}$ and ${\big (}{\tfrac {a}{p_{2}}}{\big )}$. However, it is easy to compute their product. This is known as the Jacobi symbol:
$\left({\frac {a}{N}}\right)=\left({\frac {a}{p_{1}}}\right)\left({\frac {a}{p_{2}}}\right)$
This can also be efficiently computed using the law of quadratic reciprocity for Jacobi symbols.
However, ${\big (}{\tfrac {a}{N}}{\big )}$ can not in all cases tell us whether $a$ is a quadratic residue modulo $N$ or not! More precisely, if ${\big (}{\tfrac {a}{N}}{\big )}=-1$ then $a$ is necessarily a quadratic non-residue modulo either $p_{1}$ or $p_{2}$, in which case we are done. But if ${\big (}{\tfrac {a}{N}}{\big )}=1$ then it is either the case that $a$ is a quadratic residue modulo both $p_{1}$ and $p_{2}$, or a quadratic non-residue modulo both $p_{1}$ and $p_{2}$. We cannot distinguish these cases from knowing just that ${\big (}{\tfrac {a}{N}}{\big )}=1$.
This leads to the precise formulation of the quadratic residue problem:
Problem: Given integers $a$ and $N=p_{1}p_{2}$, where $p_{1}$ and $p_{2}$ are unknown, different primes, and where ${\big (}{\tfrac {a}{N}}{\big )}=1$, determine whether $a$ is a quadratic residue modulo $N$ or not.
Distribution of residues
If $a$ is drawn uniformly at random from integers $0,\ldots ,N-1$ such that ${\big (}{\tfrac {a}{N}}{\big )}=1$, is $a$ more often a quadratic residue or a quadratic non-residue modulo $N$?
As mentioned earlier, for exactly half of the choices of $a\in \{1,\ldots ,p_{1}-1\}$, then ${\big (}{\tfrac {a}{p_{1}}}{\big )}=1$, and for the rest we have ${\big (}{\tfrac {a}{p_{1}}}{\big )}=-1$. By extension, this also holds for half the choices of $a\in \{1,\ldots ,N-1\}\setminus p_{1}\mathbb {Z} $. Similarly for $p_{2}$. From basic algebra, it follows that this partitions $(\mathbb {Z} /N\mathbb {Z} )^{\times }$ into 4 parts of equal size, depending on the sign of ${\big (}{\tfrac {a}{p_{1}}}{\big )}$ and ${\big (}{\tfrac {a}{p_{2}}}{\big )}$.
The allowed $a$ in the quadratic residue problem given as above constitute exactly those two parts corresponding to the cases ${\big (}{\tfrac {a}{p_{1}}}{\big )}={\big (}{\tfrac {a}{p_{2}}}{\big )}=1$ and ${\big (}{\tfrac {a}{p_{1}}}{\big )}={\big (}{\tfrac {a}{p_{2}}}{\big )}=-1$. Consequently, exactly half of the possible $a$ are quadratic residues and the remaining are not.
Applications
The intractability of the quadratic residuosity problem is the basis for the security of the Blum Blum Shub pseudorandom number generator. It also yields the public key Goldwasser–Micali cryptosystem.[3][4] as well as the identity based Cocks scheme.
See also
• Higher residuosity problem
References
1. Kaliski, Burt (2011). "Quadratic Residuosity Problem". Encyclopedia of Cryptography and Security: 1003. doi:10.1007/978-1-4419-5906-5_429. ISBN 978-1-4419-5905-8.
2. Adleman, L. (1980). "On Distinguishing Prime Numbers from Composite Numbers". Proceedings of the 21st IEEE Symposium on the Foundations of Computer Science (FOCS), Syracuse, N.Y. pp. 387–408. doi:10.1109/SFCS.1980.28. ISSN 0272-5428.
3. S. Goldwasser, S. Micali (1982). "Probabilistic encryption and how to play mental poker keeping secret all partial information". Proc. 14th Symposium on Theory of Computing: 365–377. doi:10.1145/800070.802212. ISBN 0897910702. S2CID 10316867.
4. S. Goldwasser, S. Micali (1984). "Probabilistic encryption". Journal of Computer and System Sciences. 28 (2): 270–299. doi:10.1016/0022-0000(84)90070-9.
Computational hardness assumptions
Number theoretic
• Integer factorization
• Phi-hiding
• RSA problem
• Strong RSA
• Quadratic residuosity
• Decisional composite residuosity
• Higher residuosity
Group theoretic
• Discrete logarithm
• Diffie-Hellman
• Decisional Diffie–Hellman
• Computational Diffie–Hellman
Pairings
• External Diffie–Hellman
• Sub-group hiding
• Decision linear
Lattices
• Shortest vector problem (gap)
• Closest vector problem (gap)
• Learning with errors
• Ring learning with errors
• Short integer solution
Non-cryptographic
• Exponential time hypothesis
• Unique games conjecture
• Planted clique conjecture
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Wikipedia
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Quadratic sieve
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties. It was invented by Carl Pomerance in 1981 as an improvement to Schroeppel's linear sieve.[1]
Basic aim
The algorithm attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n. The algorithm works in two phases: the data collection phase, where it collects information that may lead to a congruence of squares; and the data processing phase, where it puts all the data it has collected into a matrix and solves it to obtain a congruence of squares. The data collection phase can be easily parallelized to many processors, but the data processing phase requires large amounts of memory, and is difficult to parallelize efficiently over many nodes or if the processing nodes do not each have enough memory to store the whole matrix. The block Wiedemann algorithm can be used in the case of a few systems each capable of holding the matrix.
The naive approach to finding a congruence of squares is to pick a random number, square it, divide by n and hope the least non-negative remainder is a perfect square. For example, $80^{2}\equiv 441=21^{2}{\pmod {5959}}$. This approach finds a congruence of squares only rarely for large n, but when it does find one, more often than not, the congruence is nontrivial and the factorization is complete. This is roughly the basis of Fermat's factorization method.
The quadratic sieve is a modification of Dixon's factorization method.
The general running time required for the quadratic sieve (to factor an integer n) is
$e^{(1+o(1)){\sqrt {\ln n\ln \ln n}}}=L_{n}\left[1/2,1\right]$
in the L-notation.[2]
The constant e is the base of the natural logarithm.
The approach
To factorize the integer n, Fermat's method entails a search for a single number a, n1/2 < a < n−1, such that the remainder of a2 divided by n is a square. But these a are hard to find. The quadratic sieve consists of computing the remainder of a2/n for several a, then finding a subset of these whose product is a square. This will yield a congruence of squares.
For example, consider attempting to factor the number 1649. We have: $41^{2}\equiv 32,42^{2}\equiv 115,43^{2}\equiv 200{\pmod {1649}}$. None of the integers $32,115,200$ is a square, but the product $32\cdot 200=80^{2}$ is a square. We also had
$32\cdot 200\equiv 41^{2}\cdot 43^{2}=(41\cdot 43)^{2}\equiv 114^{2}{\pmod {1649}}$
since $41\cdot 43\equiv 114{\pmod {1649}}$. The observation that $32\cdot 200=80^{2}$ thus gives a congruence of squares
$114^{2}\equiv 80^{2}{\pmod {1649}}.$
Hence $114^{2}-80^{2}=(114+80)\cdot (114-80)=194\cdot 34=k\cdot 1649$ for some integer $k$. We can then factor
$1649=\gcd(194,1649)\cdot \gcd(34,1649)=97\cdot 17$
using the Euclidean algorithm to calculate the greatest common divisor.
So the problem has now been reduced to: given a set of integers, find a subset whose product is a square. By the fundamental theorem of arithmetic, any positive integer can be written uniquely as a product of prime powers. We do this in a vector format; for example, the prime-power factorization of 504 is 23325071, it is therefore represented by the exponent vector (3,2,0,1). Multiplying two integers then corresponds to adding their exponent vectors. A number is a square when its exponent vector is even in every coordinate. For example, the vectors (3,2,0,1) + (1,0,0,1) = (4,2,0,2), so (504)(14) is a square. Searching for a square requires knowledge only of the parity of the numbers in the vectors, so it is sufficient to compute these vectors mod 2: (1,0,0,1) + (1,0,0,1) = (0,0,0,0). So given a set of (0,1)-vectors, we need to find a subset which adds to the zero vector mod 2.
This is a linear algebra problem since the ring $\mathbb {Z} /2\mathbb {Z} $ can be regarded as the Galois field of order 2, that is we can divide by all non-zero numbers (there is only one, namely 1) when calculating modulo 2. It is a theorem of linear algebra that with more vectors than each vector has entries, a linear dependency always exists. It can be found by Gaussian elimination. However, simply squaring many random numbers mod n produces a very large number of different prime factors, and so very long vectors and a very large matrix. The trick is to look specifically for numbers a such that a2 mod n has only small prime factors (they are smooth numbers). They are harder to find, but using only smooth numbers keeps the vectors and matrices smaller and more tractable. The quadratic sieve searches for smooth numbers using a technique called sieving, discussed later, from which the algorithm takes its name.
The algorithm
To summarize, the basic quadratic sieve algorithm has these main steps:
1. Choose a smoothness bound B. The number π(B), denoting the number of prime numbers less than B, will control both the length of the vectors and the number of vectors needed.
2. Use sieving to locate π(B) + 1 numbers ai such that bi = (ai2 mod n) is B-smooth.
3. Factor the bi and generate exponent vectors mod 2 for each one.
4. Use linear algebra to find a subset of these vectors which add to the zero vector. Multiply the corresponding ai together and give the result mod n the name a; similarly, multiply the bi together which yields a B-smooth square b2.
5. We are now left with the equality a2 = b2 mod n from which we get two square roots of (a2 mod n), one by taking the square root in the integers of b2 namely b, and the other the a computed in step 4.
6. We now have the desired identity: $(a+b)(a-b)\equiv 0{\pmod {n}}$. Compute the GCD of n with the difference (or sum) of a and b. This produces a factor, although it may be a trivial factor (n or 1). If the factor is trivial, try again with a different linear dependency or different a.
The remainder of this article explains details and extensions of this basic algorithm.
The algorithm's pseudo code
algorithm Quadratic sieve is
Choose smoothness bound $B$
let $t=\pi (B)$
for $i\in {1,...,t+1}$ do
Choose $x\in \{0,\pm 1,\pm 2,...\}$
$a_{i}=x+\lfloor {\sqrt {n}}\rfloor $
$b_{i}=a_{i}^{2}-n$ (where $b_{i}=\prod _{j=1}^{t}p_{j}^{e_{i,j}}$)
while (check-p_t-smooth(b_i) = false) do
Let $v_{i}=(v_{i,1},v_{i,2},...,v_{i,t})=(e_{i,1}{\bmod {2}},e_{i,2}{\bmod {2}},...,e_{i,t}{\bmod {2}})$
Find $T\subseteq \{1,2,...,t+1\}:\sum _{i\in T}v_{i}=0\in \mathbb {Z} _{2}$
let $x=\prod _{i\in T}a_{i}{\bmod {n}}$
let $y=\prod _{j=1}^{t}p_{j}^{(\sum _{i\in T}e_{i,j})/2}{\bmod {n}}$
if $x\not \equiv -y{\bmod {n}}$ and $x\not \equiv y$ then
return gcd(x - y, n) , gcd(x + y, n)
else :
return to main loop.
How QS optimizes finding congruences
The quadratic sieve attempts to find pairs of integers x and y(x) (where y(x) is a function of x) satisfying a much weaker condition than x2 ≡ y2 (mod n). It selects a set of primes called the factor base, and attempts to find x such that the least absolute remainder of y(x) = x2 mod n factorizes completely over the factor base. Such y values are said to be smooth with respect to the factor base.
The factorization of a value of y(x) that splits over the factor base, together with the value of x, is known as a relation. The quadratic sieve speeds up the process of finding relations by taking x close to the square root of n. This ensures that y(x) will be smaller, and thus have a greater chance of being smooth.
$y(x)=\left(\left\lceil {\sqrt {n}}\right\rceil +x\right)^{2}-n{\hbox{ (where }}x{\hbox{ is a small integer)}}$
$y(x)\approx 2x\left\lceil {\sqrt {n}}\right\rceil $
This implies that y is on the order of 2x[√n]. However, it also implies that y grows linearly with x times the square root of n.
Another way to increase the chance of smoothness is by simply increasing the size of the factor base. However, it is necessary to find at least one smooth relation more than the number of primes in the factor base, to ensure the existence of a linear dependency.
Partial relations and cycles
Even if for some relation y(x) is not smooth, it may be possible to merge two of these partial relations to form a full one, if the two y's are products of the same prime(s) outside the factor base. [Note that this is equivalent to extending the factor base.] For example, if the factor base is {2, 3, 5, 7} and n = 91, there are partial relations:
${21^{2}\equiv 7^{1}\cdot 11{\pmod {91}}}$
${29^{2}\equiv 2^{1}\cdot 11{\pmod {91}}}$
Multiply these together:
${(21\cdot 29)^{2}\equiv 2^{1}\cdot 7^{1}\cdot 11^{2}{\pmod {91}}}$
and multiply both sides by (11−1)2 modulo 91. 11−1 modulo 91 is 58, so:
$(58\cdot 21\cdot 29)^{2}\equiv 2^{1}\cdot 7^{1}{\pmod {91}}$
$14^{2}\equiv 2^{1}\cdot 7^{1}{\pmod {91}}$
producing a full relation. Such a full relation (obtained by combining partial relations) is called a cycle. Sometimes, forming a cycle from two partial relations leads directly to a congruence of squares, but rarely.
Checking smoothness by sieving
There are several ways to check for smoothness of the ys. The most obvious is by trial division, although this increases the running time for the data collection phase. Another method that has some acceptance is the elliptic curve method (ECM). In practice, a process called sieving is typically used. If f(x) is the polynomial $f(x)=x^{2}-n$ we have
${\begin{aligned}f(x)&=x^{2}-n\\f(x+kp)&=(x+kp)^{2}-n\\&=x^{2}+2xkp+(kp)^{2}-n\\&=f(x)+2xkp+(kp)^{2}\equiv f(x){\pmod {p}}\end{aligned}}$
Thus solving f(x) ≡ 0 (mod p) for x generates a whole sequence of numbers y for which y=f(x), all of which are divisible by p. This is finding a square root modulo a prime, for which there exist efficient algorithms, such as the Shanks–Tonelli algorithm. (This is where the quadratic sieve gets its name: y is a quadratic polynomial in x, and the sieving process works like the Sieve of Eratosthenes.)
The sieve starts by setting every entry in a large array A[] of bytes to zero. For each p, solve the quadratic equation mod p to get two roots α and β, and then add an approximation to log(p) to every entry for which y(x) = 0 mod p ... that is, A[kp + α] and A[kp + β]. It is also necessary to solve the quadratic equation modulo small powers of p in order to recognise numbers divisible by small powers of a factor-base prime.
At the end of the factor base, any A[] containing a value above a threshold of roughly log(x2−n) will correspond to a value of y(x) which splits over the factor base. The information about exactly which primes divide y(x) has been lost, but it has only small factors, and there are many good algorithms for factoring a number known to have only small factors, such as trial division by small primes, SQUFOF, Pollard rho, and ECM, which are usually used in some combination.
There are many y(x) values that work, so the factorization process at the end doesn't have to be entirely reliable; often the processes misbehave on say 5% of inputs, requiring a small amount of extra sieving.
Example of basic sieve
This example will demonstrate standard quadratic sieve without logarithm optimizations or prime powers. Let the number to be factored N = 15347, therefore the ceiling of the square root of N is 124. Since N is small, the basic polynomial is enough: y(x) = (x + 124)2 − 15347.
Data collection
Since N is small, only 4 primes are necessary. The first 4 primes p for which 15347 has a square root mod p are 2, 17, 23, and 29 (in other words, 15347 is a quadratic residue modulo each of these primes). These primes will be the basis for sieving.
Now we construct our sieve $V_{X}$ of $Y(X)=(X+\lceil {\sqrt {N}}\rceil )^{2}-N=(X+124)^{2}-15347$ and begin the sieving process for each prime in the basis, choosing to sieve the first 0 ≤ X < 100 of Y(X):
${\begin{aligned}V&={\begin{bmatrix}Y(0)&Y(1)&Y(2)&Y(3)&Y(4)&Y(5)&\cdots &Y(99)\end{bmatrix}}\\&={\begin{bmatrix}29&278&529&782&1037&1294&\cdots &34382\end{bmatrix}}\end{aligned}}$
The next step is to perform the sieve. For each p in our factor base $\lbrace 2,17,23,29\rbrace $ solve the equation
$Y(X)\equiv (X+\lceil {\sqrt {N}}\rceil )^{2}-N\equiv 0{\pmod {p}}$
to find the entries in the array V which are divisible by p.
For $p=2$ solve $(X+124)^{2}-15347\equiv 0{\pmod {2}}$ to get the solution $X\equiv {\sqrt {15347}}-124\equiv 1{\pmod {2}}$.
Thus, starting at X=1 and incrementing by 2, each entry will be divisible by 2. Dividing each of those entries by 2 yields
$V={\begin{bmatrix}29&139&529&391&1037&647&\cdots &17191\end{bmatrix}}$
Similarly for the remaining primes p in $\lbrace 17,23,29\rbrace $ the equation$X\equiv {\sqrt {15347}}-124{\pmod {p}}$ is solved. Note that for every p > 2, there will be 2 resulting linear equations due to there being 2 modular square roots.
${\begin{aligned}X&\equiv {\sqrt {15347}}-124&\equiv 8-124&\equiv 3{\pmod {17}}\\&&\equiv 9-124&\equiv 4{\pmod {17}}\\X&\equiv {\sqrt {15347}}-124&\equiv 11-124&\equiv 2{\pmod {23}}\\&&\equiv 12-124&\equiv 3{\pmod {23}}\\X&\equiv {\sqrt {15347}}-124&\equiv 8-124&\equiv 0{\pmod {29}}\\&&\equiv 21-124&\equiv 13{\pmod {29}}\\\end{aligned}}$
Each equation $X\equiv a{\pmod {p}}$ results in $V_{x}$ being divisible by p at x=a and each pth value beyond that. Dividing V by p at a, a+p, a+2p, a+3p, etc., for each prime in the basis finds the smooth numbers which are products of unique primes (first powers).
$V={\begin{bmatrix}1&139&23&1&61&647&\cdots &17191\end{bmatrix}}$
Any entry of V that equals 1 corresponds to a smooth number. Since $V_{0}$, $V_{3}$, and $V_{71}$ equal one, this corresponds to:
X + 124Yfactors
124 29 20 • 170 • 230 • 291
127 782 21 • 171 • 231 • 290
195 22678 21 • 171 • 231 • 291
Matrix processing
Since smooth numbers Y have been found with the property $Y\equiv Z^{2}{\pmod {N}}$, the remainder of the algorithm follows equivalently to any other variation of Dixon's factorization method.
Writing the exponents of the product of a subset of the equations
${\begin{aligned}29&=2^{0}\cdot 17^{0}\cdot 23^{0}\cdot 29^{1}\\782&=2^{1}\cdot 17^{1}\cdot 23^{1}\cdot 29^{0}\\22678&=2^{1}\cdot 17^{1}\cdot 23^{1}\cdot 29^{1}\\\end{aligned}}$
as a matrix ${\pmod {2}}$ yields:
$S\cdot {\begin{bmatrix}0&0&0&1\\1&1&1&0\\1&1&1&1\end{bmatrix}}\equiv {\begin{bmatrix}0&0&0&0\end{bmatrix}}{\pmod {2}}$
A solution to the equation is given by the left null space, simply
$S={\begin{bmatrix}1&1&1\end{bmatrix}}$
Thus the product of all 3 equations yields a square (mod N).
$29\cdot 782\cdot 22678=22678^{2}$
and
$124^{2}\cdot 127^{2}\cdot 195^{2}=3070860^{2}$
So the algorithm found
$22678^{2}\equiv 3070860^{2}{\pmod {15347}}$
Testing the result yields GCD(3070860 - 22678, 15347) = 103, a nontrivial factor of 15347, the other being 149.
This demonstration should also serve to show that the quadratic sieve is only appropriate when n is large. For a number as small as 15347, this algorithm is overkill. Trial division or Pollard rho could have found a factor with much less computation.
Multiple polynomials
In practice, many different polynomials are used for y, since only one polynomial will not typically provide enough (x, y) pairs that are smooth over the factor base. The polynomials used must have a special form, since they need to be squares modulo n. The polynomials must all have a similar form to the original y(x) = x2 − n:
$y(x)=(Ax+B)^{2}-n\qquad A,B\in \mathbb {Z} $
Assuming $B^{2}-n$ is a multiple of A, so that $B^{2}-n=AC$ the polynomial y(x) can be written as $y(x)=A\cdot (Ax^{2}+2Bx+C)$. If A is a square, then only the factor $(Ax^{2}+2Bx+C)$ has to be considered.
This approach (called MPQS, Multiple Polynomial Quadratic Sieve) is ideally suited for parallelization, since each processor involved in the factorization can be given n, the factor base and a collection of polynomials, and it will have no need to communicate with the central processor until it is finished with its polynomials.
Large primes
One large prime
If, after dividing by all the factors less than A, the remaining part of the number (the cofactor) is less than A2, then this cofactor must be prime. In effect, it can be added to the factor base, by sorting the list of relations into order by cofactor. If y(a) = 7*11*23*137 and y(b) = 3*5*7*137, then y(a)y(b) = 3*5*11*23 * 72 * 1372. This works by reducing the threshold of entries in the sieving array above which a full factorization is performed.
More large primes
Reducing the threshold even further, and using an effective process for factoring y(x) values into products of even relatively large primes - ECM is superb for this - can find relations with most of their factors in the factor base, but with two or even three larger primes. Cycle finding then allows combining a set of relations sharing several primes into a single relation.
Parameters from realistic example
To illustrate typical parameter choices for a realistic example on a real implementation including the multiple polynomial and large prime optimizations, the tool msieve was run on a 267-bit semiprime, producing the following parameters:
• Trial factoring cutoff: 27 bits
• Sieve interval (per polynomial): 393216 (12 blocks of size 32768)
• Smoothness bound: 1300967 (50294 primes)
• Number of factors for polynomial A coefficients: 10 (see Multiple polynomials above)
• Large prime bound: 128795733 (26 bits) (see Large primes above)
• Smooth values found: 25952 by sieving directly, 24462 by combining numbers with large primes
• Final matrix size: 50294 × 50414, reduced by filtering to 35750 × 35862
• Nontrivial dependencies found: 15
• Total time (on a 1.6 GHz UltraSPARC III): 35 min 39 seconds
• Maximum memory used: 8 MB
Factoring records
Until the discovery of the number field sieve (NFS), QS was the asymptotically fastest known general-purpose factoring algorithm. Now, Lenstra elliptic curve factorization has the same asymptotic running time as QS (in the case where n has exactly two prime factors of equal size), but in practice, QS is faster since it uses single-precision operations instead of the multi-precision operations used by the elliptic curve method.
On April 2, 1994, the factorization of RSA-129 was completed using QS. It was a 129-digit number, the product of two large primes, one of 64 digits and the other of 65. The factor base for this factorization contained 524339 primes. The data collection phase took 5000 MIPS-years, done in distributed fashion over the Internet. The data collected totaled 2GB. The data processing phase took 45 hours on Bellcore's (now Telcordia Technologies) MasPar (massively parallel) supercomputer. This was the largest published factorization by a general-purpose algorithm, until NFS was used to factor RSA-130, completed April 10, 1996. All RSA numbers factored since then have been factored using NFS.
The current QS factorization record is the 140 digit (463 bit) RSA-140, which was factored by Patrick Konsor in June 2020 using approximately 6,000 core hours over 6 days.[3]
Implementations
• PPMPQS and PPSIQS
• mpqs
• SIMPQS is a fast implementation of the self-initialising multiple polynomial quadratic sieve written by William Hart. It provides support for the large prime variant and uses Jason Papadopoulos' block Lanczos code for the linear algebra stage. SIMPQS is accessible as the qsieve command in the SageMath computer algebra package or can be downloaded in source form. SIMPQS is optimized for use on Athlon and Opteron machines, but will operate on most common 32- and 64-bit architectures. It is written entirely in C.
• a factoring applet by Dario Alpern, that uses the quadratic sieve if certain conditions are met.
• The PARI/GP computer algebra package includes an implementation of the self-initialising multiple polynomial quadratic sieve implementing the large prime variant. It was adapted by Thomas Papanikolaou and Xavier Roblot from a sieve written for the LiDIA project. The self initialisation scheme is based on an idea from the thesis of Thomas Sosnowski.
• A variant of the quadratic sieve is available in the MAGMA computer algebra package. It is based on an implementation of Arjen Lenstra from 1995, used in his "factoring by email" program.
• msieve, an implementation of the multiple polynomial quadratic sieve with support for single and double large primes, written by Jason Papadopoulos. Source code and a Windows binary are available.
• YAFU, written by Ben Buhrow, is probably the fastest available implementation of the quadratic sieve. For example, RSA-100 was factored in less than 15 minutes on 4 cores of a 2.5 GHz Xeon 6248 CPU. All of the critical subroutines make use of AVX2 or AVX-512 SIMD instructions for AMD or Intel processors. It uses Jason Papadopoulos' block Lanczos code. Source code and binaries for Windows and Linux are available.
• Ariel, a simple Java implementation of the quadratic sieve for didactic purposes.
• The java-math-library contains probably the fastest quadratic sieve written in Java (the successor of PSIQS 4.0).
• Java QS, an open source Java project containing basic implementation of QS. Released at February 4, 2016 by Ilya Gazman
• C Quadratic Sieve, a factorizer written entirely in C containing implementation of self-initialising Quadratic Sieve. The project is inspired by a William Hart's FLINT factorizer. The source released in 2022 by Michel Leonard does not rely on external library, it is capable of factoring 240-bit numbers in a minute and 300-bit numbers in 2 hours.
• The RcppBigIntAlgos package by Joseph Wood, provides an efficient implementation of the multiple polynomial quadratic sieve for the R programming language. It is written in C++ and is capable of comfortably factoring 100 digit semiprimes. For example, a 300-bit semiprime (91 digits) was factored in 1 hour and the RSA-100 was factored in under 10 hours on a MacBook Air with an Apple M2 processor.
See also
Wikiversity has learning resources about Quadratic Sieve
• Lenstra elliptic curve factorization
• primality test
References
1. Carl Pomerance, Analysis and Comparison of Some Integer Factoring Algorithms, in Computational Methods in Number Theory, Part I, H.W. Lenstra, Jr. and R. Tijdeman, eds., Math. Centre Tract 154, Amsterdam, 1982, pp 89-139.
2. Pomerance, Carl (December 1996). "A Tale of Two Sieves" (PDF). Notices of the AMS. Vol. 43, no. 12. pp. 1473–1485.
3. "Useless Accomplishment: RSA-140 Factorization with Quadratic Sieve - mersenneforum.org". www.mersenneforum.org. Retrieved 2020-07-07.
• Richard Crandall and Carl Pomerance (2001). Prime Numbers: A Computational Perspective (1st ed.). Springer. ISBN 0-387-94777-9. Section 6.1: The quadratic sieve factorization method, pp. 227–244.
• Samuel S. Wagstaff, Jr. (2013). The Joy of Factoring. Providence, RI: American Mathematical Society. pp. 195–202. ISBN 978-1-4704-1048-3.
Other external links
• Reference paper "The Quadratic Sieve Factoring Algorithm" by Eric Landquist
Number-theoretic algorithms
Primality tests
• AKS
• APR
• Baillie–PSW
• Elliptic curve
• Pocklington
• Fermat
• Lucas
• Lucas–Lehmer
• Lucas–Lehmer–Riesel
• Proth's theorem
• Pépin's
• Quadratic Frobenius
• Solovay–Strassen
• Miller–Rabin
Prime-generating
• Sieve of Atkin
• Sieve of Eratosthenes
• Sieve of Pritchard
• Sieve of Sundaram
• Wheel factorization
Integer factorization
• Continued fraction (CFRAC)
• Dixon's
• Lenstra elliptic curve (ECM)
• Euler's
• Pollard's rho
• p − 1
• p + 1
• Quadratic sieve (QS)
• General number field sieve (GNFS)
• Special number field sieve (SNFS)
• Rational sieve
• Fermat's
• Shanks's square forms
• Trial division
• Shor's
Multiplication
• Ancient Egyptian
• Long
• Karatsuba
• Toom–Cook
• Schönhage–Strassen
• Fürer's
Euclidean division
• Binary
• Chunking
• Fourier
• Goldschmidt
• Newton-Raphson
• Long
• Short
• SRT
Discrete logarithm
• Baby-step giant-step
• Pollard rho
• Pollard kangaroo
• Pohlig–Hellman
• Index calculus
• Function field sieve
Greatest common divisor
• Binary
• Euclidean
• Extended Euclidean
• Lehmer's
Modular square root
• Cipolla
• Pocklington's
• Tonelli–Shanks
• Berlekamp
• Kunerth
Other algorithms
• Chakravala
• Cornacchia
• Exponentiation by squaring
• Integer square root
• Integer relation (LLL; KZ)
• Modular exponentiation
• Montgomery reduction
• Schoof
• Trachtenberg system
• Italics indicate that algorithm is for numbers of special forms
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Quadratically closed field
In mathematics, a quadratically closed field is a field in which every element has a square root.[1][2]
Examples
• The field of complex numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed.
• The field of real numbers is not quadratically closed as it does not contain a square root of −1.
• The union of the finite fields $\mathbb {F} _{5^{2^{n}}}$ for n ≥ 0 is quadratically closed but not algebraically closed.[3]
• The field of constructible numbers is quadratically closed but not algebraically closed.[4]
Properties
• A field is quadratically closed if and only if it has universal invariant equal to 1.
• Every quadratically closed field is a Pythagorean field but not conversely (for example, R is Pythagorean); however, every non-formally real Pythagorean field is quadratically closed.[2]
• A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to Z under the dimension mapping.[3]
• A formally real Euclidean field E is not quadratically closed (as −1 is not a square in E) but the quadratic extension E(√−1) is quadratically closed.[4]
• Let E/F be a finite extension where E is quadratically closed. Either −1 is a square in F and F is quadratically closed, or −1 is not a square in F and F is Euclidean. This "going-down theorem" may be deduced from the Diller–Dress theorem.[5]
Quadratic closure
A quadratic closure of a field F is a quadratically closed field containing F which embeds in any quadratically closed field containing F. A quadratic closure for any given F may be constructed as a subfield of the algebraic closure Falg of F, as the union of all iterated quadratic extensions of F in Falg.[4]
Examples
• The quadratic closure of R is C.[4]
• The quadratic closure of $\mathbb {F} _{5}$ is the union of the $\mathbb {F} _{5^{2^{n}}}$.[4]
• The quadratic closure of Q is the field of complex constructible numbers.
References
1. Lam (2005) p. 33
2. Rajwade (1993) p. 230
3. Lam (2005) p. 34
4. Lam (2005) p. 220
5. Lam (2005) p.270
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Rajwade, A. R. (1993). Squares. London Mathematical Society Lecture Note Series. Vol. 171. Cambridge University Press. ISBN 0-521-42668-5. Zbl 0785.11022.
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Quadratically constrained quadratic program
In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form
${\begin{aligned}&{\text{minimize}}&&{\tfrac {1}{2}}x^{\mathrm {T} }P_{0}x+q_{0}^{\mathrm {T} }x\\&{\text{subject to}}&&{\tfrac {1}{2}}x^{\mathrm {T} }P_{i}x+q_{i}^{\mathrm {T} }x+r_{i}\leq 0\quad {\text{for }}i=1,\dots ,m,\\&&&Ax=b,\end{aligned}}$
where P0, …, Pm are n-by-n matrices and x ∈ Rn is the optimization variable.
If P0, …, Pm are all positive semidefinite, then the problem is convex. If these matrices are neither positive nor negative semidefinite, the problem is non-convex. If P1, … ,Pm are all zero, then the constraints are in fact linear and the problem is a quadratic program.
Hardness
Solving the general case is an NP-hard problem. To see this, note that the two constraints x1(x1 − 1) ≤ 0 and x1(x1 − 1) ≥ 0 are equivalent to the constraint x1(x1 − 1) = 0, which is in turn equivalent to the constraint x1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard.
Relaxation
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.[1]
Nonconvex QCQPs with non-positive off-diagonal elements can be exactly solved by the SDP or SOCP relaxations,[2] and there are polynomial-time-checkable sufficient conditions for SDP relaxations of general QCQPs to be exact.[3] Moreover, it was shown that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables.[3]
Semidefinite programming
When P0, …, Pm are all positive-definite matrices, the problem is convex and can be readily solved using interior point methods, as done with semidefinite programming.
Example
Max Cut is a problem in graph theory, which is NP-hard. Given a graph, the problem is to divide the vertices in two sets, so that as many edges as possible go from one set to the other. Max Cut can be formulated as a QCQP, and SDP relaxation of the dual provides good lower bounds.
Solvers and scripting (programming) languages
Name Brief info
Artelys KnitroKnitro is a solver specialized in nonlinear optimization, but also solves linear programming problems, quadratic programming problems, second-order cone programming, systems of nonlinear equations, and problems with equilibrium constraints.
FICO XpressA commercial optimization solver for linear programming, non-linear programming, mixed integer linear programming, convex quadratic programming, convex quadratically constrained quadratic programming, second-order cone programming and their mixed integer counterparts.
AMPL
CPLEXPopular solver with an API for several programming languages. Free for academics.
MOSEKA solver for large scale optimization with API for several languages (C++,java,.net, Matlab and python)
TOMLABSupports global optimization, integer programming, all types of least squares, linear, quadratic and unconstrained programming for MATLAB. TOMLAB supports solvers like CPLEX, SNOPT and KNITRO.
Wolfram MathematicaAble to solve QCPQ type of problems using functions like Minimize.
References
1. Kimizuka, Masaki; Kim, Sunyoung; Yamashita, Makoto (2019). "Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods". Journal of Global Optimization. 75 (3): 631–654. doi:10.1007/s10898-019-00795-w. ISSN 0925-5001. S2CID 254701008.
2. Kim, Sunyoung; Kojima, Masakazu (2003). "Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations". Computational Optimization and Applications. 26 (2): 143–154. doi:10.1023/A:1025794313696. S2CID 1241391.
3. Burer, Samuel; Ye, Yinyu (2019-02-04). "Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs". Mathematical Programming. 181: 1–17. arXiv:1802.02688. doi:10.1007/s10107-019-01367-2. ISSN 0025-5610. S2CID 254143721.
• Boyd, Stephen; Lieven Vandenberghe (2004). Convex Optimization. Cambridge: Cambridge University Press. ISBN 978-0-521-83378-3.
Further reading
In statistics
• Albers C. J., Critchley F., Gower, J. C. (2011). "Quadratic Minimisation Problems in Statistics" (PDF). Journal of Multivariate Analysis. 102 (3): 698–713. doi:10.1016/j.jmva.2009.12.018. hdl:11370/6295bde7-4de1-48c2-a30b-055eff924f3e.{{cite journal}}: CS1 maint: multiple names: authors list (link)
External links
• NEOS Optimization Guide: Quadratic Constrained Quadratic Programming
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Quadratrix
In geometry, a quadratrix (from Latin quadrator 'squarer') is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.
Quadratrix of Dinostratus
The quadratrix of Dinostratus (also called the quadratrix of Hippias) was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus, in his Collections, treats its history, and gives two methods by which it can be generated.
1. Let a helix be drawn on a right circular cylinder; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis. The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.
2. A right cylinder having for its base an Archimedean spiral is intersected by a right circular cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.
Another construction is as follows. DAB is a quadrant in which the line DA and the arc DB are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to AB and through the corresponding points on the radius DA. The locus of these intersections is the quadratrix.
Letting A be the origin of the Cartesian coordinate system, D be the point (a, 0), a units from the origin along the x-axis, and B be the point (0, a), a units from the origin along the y-axis, the curve itself can be expressed by the equation[1]
$y=x\cot \left({\frac {\pi x}{2a}}\right).$
Because the cotangent function is invariant under negation of its argument, and has a simple pole at each multiple of π, the quadratrix has reflection symmetry across the y-axis, and similarly has a pole for each value of x of the form x = 2na, for integer values of n, except at x = 0 where the pole in the cotangent is canceled by the factor of x in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the y-axis has y = 2a/π; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of 1/π, leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge. An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and straightedge: doubling the cube and trisecting an angle.
Quadratrix of Tschirnhaus
The quadratrix of Tschirnhaus[2] is constructed by dividing the arc and radius of a quadrant in the same number of equal parts as before. The mutual intersections of the lines drawn from the points of division of the arc parallel to DA, and the lines drawn parallel to AB through the points of division of DA, are points on the quadratrix. The Cartesian equation is $y=a\cos \!{\big (}{\tfrac {\pi x}{2a}}{\big )}$. The curve is periodic, and cuts the x-axis at the points $x=(2n-1)a$, $n$ being an integer; the maximum values of $y$ are $a$. Its properties are similar to those of the quadratrix of Dinostratus.
Other quadratrices
Other curves that have historically been used to square the circle include:
• Archimedean spiral
• Ozanam's quadratrix
• Cochleoid
References
• This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Quadratrix". Encyclopædia Britannica. Vol. 22 (11th ed.). Cambridge University Press. p. 706.
1. "Dinostratus quadratrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. See definition and drawing in the following online source: Hutton C. (1815). A Philosophical and Mathematical Dictionary Containing... Memoirs of the Lives and Writings of the Most Eminent Authors. Vol. 2. London. pp. 271–272.
External links
Wikimedia Commons has media related to Quadratrix.
• Quadratrix of Hippias Archived 2012-02-04 at the Wayback Machine at the MacTutor archive.
• Hippias' Quadratrix at Convergence (MAA periodical)
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Quadrature domains
In the branch of mathematics called potential theory, a quadrature domain in two dimensional real Euclidean space is a domain D (an open connected set) together with a finite subset {z1, …, zk} of D such that, for every function u harmonic and integrable over D with respect to area measure, the integral of u with respect to this measure is given by a "quadrature formula"; that is,
$\iint _{D}u\,dxdy=\sum _{j=1}^{k}c_{j}u(z_{j}),$
where the cj are nonzero complex constants independent of u.
The most obvious example is when D is a circular disk: here k = 1, z1 is the center of the circle, and c1 equals the area of D. That quadrature formula expresses the mean value property of harmonic functions with respect to disks.
It is known that quadrature domains exist for all values of k. There is an analogous definition of quadrature domains in Euclidean space of dimension d larger than 2. There is also an alternative, electrostatic interpretation of quadrature domains: a domain D is a quadrature domain if a uniform distribution of electric charge on D creates the same electrostatic field outside D as does a k-tuple of point charges at the points z1, …, zk.
Quadrature domains and numerous generalizations thereof (e.g., replace area measure by length measure on the boundary of D) have in recent years been encountered in various connections such as inverse problems of Newtonian gravitation, Hele-Shaw flows of viscous fluids, and purely mathematical isoperimetric problems, and interest in them seems to be steadily growing. They were the subject of an international conference at the University of California at Santa Barbara in 2003 and the state of the art as of that date can be seen in the proceedings of that conference, published by Birkhäuser Verlag.
References
• Ebenfelt, Peter (2005). Quadrature Domains and Their Applications: The Harold S. Shapiro Anniversary Volume. Birkhäuser. ISBN 3-7643-7145-5. Retrieved 2007-04-11.
• Aharonov, Dov; Shapiro, Harold S. (1976). "Domains on which analytic functions satisfy quadrature identities". Journal d'Analyse Mathématique. 30: 39–73. doi:10.1007/BF02786704.
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Quadrature filter
In signal processing, a quadrature filter $q(t)$ is the analytic representation of the impulse response $f(t)$ of a real-valued filter:
$q(t)=f_{a}(t)=\left(\delta (t)+j\delta (jt)\right)*f(t)$
If the quadrature filter $q(t)$ is applied to a signal $s(t)$, the result is
$h(t)=(q*s)(t)=\left(\delta (t)+j\delta (jt)\right)*f(t)*s(t)$
which implies that $h(t)$ is the analytic representation of $(f*s)(t)$.
Since $q$ is an analytic signal, it is either zero or complex-valued. In practice, therefore, $q$ is often implemented as two real-valued filters, which correspond to the real and imaginary parts of the filter, respectively.
An ideal quadrature filter cannot have a finite support. It has single sided support, but by choosing the (analog) function $f(t)$ carefully, it is possible to design quadrature filters which are localized such that they can be approximated by means of functions of finite support. A digital realization without feedback (FIR) has finite support.
Applications
This construction will simply assemble an analytic signal with a starting point to finally create a causal signal with finite energy. The two Delta Distributions will perform this operation. This will impose an additional constraint on the filter.
Single frequency signals
For single frequency signals (in practice narrow bandwidth signals) with frequency $\omega $ the magnitude of the response of a quadrature filter equals the signal's amplitude A times the frequency function of the filter at frequency $\omega $.
$h(t)=(s*q)(t)={\frac {1}{\pi }}\int _{0}^{\infty }S(u)Q(u)e^{iut}du={\frac {1}{\pi }}\int _{0}^{\infty }A\pi \delta (u-\omega )Q(u)e^{iut}du=$
$=A\int _{0}^{\infty }\delta (u-\omega )Q(u)e^{iut}du=AQ(\omega )e^{i\omega t}$
$|h(t)|=A|Q(\omega )|$
This property can be useful when the signal s is a narrow-bandwidth signal of unknown frequency. By choosing a suitable frequency function Q of the filter, we may generate known functions of the unknown frequency $\omega $ which then can be estimated.
See also
• Analytic signal
• Hilbert transform
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Lune of Hippocrates
In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It was the first curved figure to have its exact area calculated mathematically.[1]
History
Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given circle.[2][3] He proved that the lune bounded by the arcs labeled E and F in the figure has the same area as triangle ABO. This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles. Heath concludes that, in proving his result, Hippocrates was also the first to prove that the area of a circle is proportional to the square of its diameter.[2]
Hippocrates' book on geometry in which this result appears, Elements, has been lost, but may have formed the model for Euclid's Elements.[3] Hippocrates' proof was preserved through the History of Geometry compiled by Eudemus of Rhodes, which has also not survived, but which was excerpted by Simplicius of Cilicia in his commentary on Aristotle's Physics.[2][4]
Not until 1882, with Ferdinand von Lindemann's proof of the transcendence of π, was squaring the circle proved to be impossible.[5]
Proof
Hippocrates' result can be proved as follows: The center of the circle on which the arc AEB lies is the point D, which is the midpoint of the hypotenuse of the isosceles right triangle ABO. Therefore, the diameter AC of the larger circle ABC is ${\sqrt {2}}$ times the diameter of the smaller circle on which the arc AEB lies. Consequently, the smaller circle has half the area of the larger circle, and therefore the quarter circle AFBOA is equal in area to the semicircle AEBDA. Subtracting the crescent-shaped area AFBDA from the quarter circle gives triangle ABO and subtracting the same crescent from the semicircle gives the lune. Since the triangle and lune are both formed by subtracting equal areas from equal area, they are themselves equal in area.[2][6]
Generalizations
Using a similar proof to the one above, the Arab mathematician Hasan Ibn al-Haytham (Latinized name Alhazen, c. 965 – c. 1040) showed that where two lunes are formed, on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle of the triangle, then the areas of these two lunes added together are equal to the area of the triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen.[7][8] The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle.[9]
All lunes constructable by compass and straight-edge can be specified by the two angles formed by the inner and outer arcs on their respective circles; in this notation, for instance, the lune of Hippocrates would have the inner and outer angles (90°, 180°) with ratio 1:2. Hippocrates found two other squarable concave lunes, with angles approximately (107.2°, 160.9°) with ratio 2:3 and (68.5°, 205.6°) with ratio 1:3. Two more squarable concave lunes, with angles approximately (46.9°, 234.4°) with ratio 1:5 and (100.8°, 168.0°) with ratio 3:5 were found in 1766 by Martin Johan Wallenius and again in 1840 by Thomas Clausen. In the mid-20th century, two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square. As Chebotaryov and Dorodnov showed, these five pairs of angles give the only constructible squarable lunes; in particular, there are no other constructible squarable lunes.[1][8]
References
1. Postnikov, M. M. (2000), "The problem of squarable lunes", American Mathematical Monthly, 107 (7): 645–651, doi:10.2307/2589121, JSTOR 2589121. Translated from Postnikov's 1963 Russian book on Galois theory.
2. Heath, Thomas L. (2003), A Manual of Greek Mathematics, Courier Dover Publications, pp. 121–132, ISBN 0-486-43231-9.
3. "Hippocrates of Chios", Encyclopædia Britannica, 2012, retrieved 2012-01-12.
4. O'Connor, John J.; Robertson, Edmund F., "Hippocrates of Chios", MacTutor History of Mathematics Archive, University of St Andrews
5. Jacobs, Konrad (1992), "2.1 Squaring the Circle", Invitation to Mathematics, Princeton University Press, pp. 11–13, ISBN 978-0-691-02528-5.
6. Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1988), "4-2 Hippocrates of Chios and the quadrature of lunes", The Historical Roots of Elementary Mathematics, Courier Dover Publications, pp. 90–91, ISBN 0-486-25563-8.
7. Hippocrates' Squaring of the Lune at cut-the-knot, accessed 2012-01-12.
8. Alsina, Claudi; Nelsen, Roger B. (2010), "9.1 Squarable lunes", Charming Proofs: A Journey into Elegant Mathematics, Dolciani mathematical expositions, vol. 42, Mathematical Association of America, pp. 137–144, ISBN 978-0-88385-348-1.
9. Anglin, W. S. (1994), "Hippocrates and the Lunes", Mathematics, a Concise History and Philosophy, Springer, pp. 51–53, ISBN 0-387-94280-7.
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Quadrature of the Parabola
Quadrature of the Parabola (Greek: Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a line) is ${\tfrac {4}{3}}$ that of a certain inscribed triangle.
It is one of the best-known works of Archimedes, in particular for its ingenious use of the method of exhaustion and in the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression.[1] He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of a reductio ad absurdum argument in ancient Greek mathematics, and Archimedes' solution remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri's quadrature formula.[2]
Main theorem
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord. Proposition 1 of the work states that a line from the third vertex drawn parallel to the axis divides the chord into equal segments. The main theorem claims that the area of the parabolic segment is ${\tfrac {4}{3}}$ that of the inscribed triangle.
Structure of the text
Conic sections such as the parabola were already well known in Archimedes' time thanks to Menaechmus a century earlier. However, before the advent of the differential and integral calculus, there were no easy means to find the area of a conic section. Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord.[3]
Archimedes gives two proofs of the main theorem: one using abstract mechanics and the other one by pure geometry. In the first proof, Archimedes considers a lever in equilibrium under the action of gravity, with weighted segments of a parabola and a triangle suspended along the arms of a lever at specific distances from the fulcrum.[4] When the center of gravity of the triangle is known, the equilibrium of the lever yields the area of the parabola in terms of the area of the triangle which has the same base and equal height.[5] Archimedes here deviates from the procedure found in On the Equilibrium of Planes in that he has the centers of gravity at a level below that of the balance.[6] The second and more famous proof uses pure geometry, particularly the sum of a geometric series.
Of the twenty-four propositions, the first three are quoted without proof from Euclid's Elements of Conics (a lost work by Euclid on conic sections). Propositions 4 and 5 establish elementary properties of the parabola. Propositions 6–17 give the mechanical proof of the main theorem; propositions 18–24 present the geometric proof.
Geometric proof
Dissection of the parabolic segment
The main idea of the proof is the dissection of the parabolic segment into infinitely many triangles, as shown in the figure to the right. Each of these triangles is inscribed in its own parabolic segment in the same way that the blue triangle is inscribed in the large segment.
Areas of the triangles
In propositions eighteen through twenty-one, Archimedes proves that the area of each green triangle is ${\tfrac {1}{8}}$ the area of the blue triangle, so that both green triangles together sum to ${\tfrac {1}{4}}$ the area of the blue triangle. From a modern point of view, this is because the green triangle has ${\tfrac {1}{2}}$ the width and ${\tfrac {1}{4}}$ the height of the blue triangle:[7]
Following the same argument, each of the $4$ yellow triangles has ${\tfrac {1}{8}}$ the area of a green triangle or ${\tfrac {1}{64}}$ the area of the blue triangle, summing to ${\tfrac {4}{64}}={\tfrac {1}{16}}$ the area of the blue triangle; each of the $2^{3}=8$ red triangles has ${\tfrac {1}{8}}$ the area of a yellow triangle, summing to ${\tfrac {2^{3}}{8^{3}}}={\tfrac {1}{64}}$ the area of the blue triangle; etc. Using the method of exhaustion, it follows that the total area of the parabolic segment is given by
${\text{Area}}\;=\;T\,+\,{\frac {1}{4}}T\,+\,{\frac {1}{4^{2}}}T\,+\,{\frac {1}{4^{3}}}T\,+\,\cdots .$
Here T represents the area of the large blue triangle, the second term represents the total area of the two green triangles, the third term represents the total area of the four yellow triangles, and so forth. This simplifies to give
${\text{Area}}\;=\;\left(1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots \right)T.$
Sum of the series
To complete the proof, Archimedes shows that
$1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots \;=\;{\frac {4}{3}}.$
The formula above is a geometric series—each successive term is one fourth of the previous term. In modern mathematics, that formula is a special case of the sum formula for a geometric series.
Archimedes evaluates the sum using an entirely geometric method,[8] illustrated in the adjacent picture. This picture shows a unit square which has been dissected into an infinity of smaller squares. Each successive purple square has one fourth the area of the previous square, with the total purple area being the sum
${\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots .$
However, the purple squares are congruent to either set of yellow squares, and so cover ${\tfrac {1}{3}}$ of the area of the unit square. It follows that the series above sums to ${\tfrac {4}{3}}$ (since $1+{\tfrac {1}{3}}={\tfrac {4}{3}}$).
See also
Wikimedia Commons has media related to Quadrature of the Parabola.
• History of calculus
Notes
1. Swain, Gordon; Dence, Thomas (1998). "Archimedes' Quadrature of the Parabola Revisited". Mathematics Magazine. 71 (2): 123–130. doi:10.2307/2691014. ISSN 0025-570X. JSTOR 2691014.
2. Cusick, Larry W. (2008). "Archimedean Quadrature Redux". Mathematics Magazine. 81 (2): 83–95. doi:10.1080/0025570X.2008.11953535. ISSN 0025-570X. JSTOR 27643090. S2CID 126360876.
3. Towne, R. (2018). "Archimedes in the Classroom". Master's Thesis. John Carroll University.
4. "Quadrature of the parabola, Introduction". web.calstatela.edu. Retrieved 2021-07-03.
5. "The Illustrated Method of Archimedes". Scribd. Retrieved 2021-07-03.
6. Dijksterhuis, E. J. (1987). "Quadrature of the Parabola". Archimedes. pp. 336–345.{{cite web}}: CS1 maint: url-status (link)
7. The green triangle has ${\tfrac {1}{2}}$ the width of blue triangle by construction. The statement about the height follows from the geometric properties of a parabola, and is easy to prove using modern analytic geometry.
8. Strictly speaking, Archimedes evaluates the partial sums of this series, and uses the Archimedean property to argue that the partial sums become arbitrarily close to ${\tfrac {4}{3}}$. This is logically equivalent to the modern idea of summing an infinite series.
Further reading
• Ajose, Sunday and Roger Nelsen (June 1994). "Proof without Words: Geometric Series". Mathematics Magazine. 67 (3): 230. doi:10.2307/2690617. JSTOR 2690617.
• Ancora, Luciano (2014). "Quadrature of the parabola with the square pyramidal number". Archimede. 66 (3).
• Bressoud, David M. (2006). A Radical Approach to Real Analysis (2nd ed.). Mathematical Association of America. ISBN 0-88385-747-2..
• Dijksterhuis, E.J. (1987) "Archimedes", Princeton U. Press ISBN 0-691-08421-1
• Edwards Jr., C. H. (1994). The Historical Development of the Calculus (3rd ed.). Springer. ISBN 0-387-94313-7..
• Heath, Thomas L. (2011). The Works of Archimedes (2nd ed.). CreateSpace. ISBN 978-1-4637-4473-1.
• Simmons, George F. (2007). Calculus Gems. Mathematical Association of America. ISBN 978-0-88385-561-4..
• Stein, Sherman K. (1999). Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 0-88385-718-9.
• Stillwell, John (2004). Mathematics and its History (2nd ed.). Springer. ISBN 0-387-95336-1..
• Swain, Gordon and Thomas Dence (April 1998). "Archimedes' Quadrature of the Parabola Revisited". Mathematics Magazine. 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014.
• Wilson, Alistair Macintosh (1995). The Infinite in the Finite. Oxford University Press. ISBN 0-19-853950-9..
External links
Look up quadrature in Wiktionary, the free dictionary.
• Casselman, Bill. "Archimedes' quadrature of the parabola". Archived from the original on 2012-02-04. Full text, as translated by T.L. Heath.
• Xavier University Department of Mathematics and Computer Science. "Archimedes of Syracuse". Archived from the original on 2016-01-13.. Text of propositions 1–3 and 20–24, with commentary.
• http://planetmath.org/ArchimedesCalculus
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Quadray coordinates
Quadray coordinates, also known as caltrop, tetray or Chakovian coordinates, were developed by Darrel Jarmusch and others, as another take on simplicial coordinates, a coordinate system using a simplex or tetrahedron as its basis polyhedron.[1]
Geometric definition
The four basis (but not necessarily unit) vectors stem from the center of a regular tetrahedron and go to its four corners. Their coordinate addresses are (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) respectively. These may be positively scaled without rotation (e.g. negation) and linearly combined to span conventional XYZ space, with at least one of the four coordinates unneeded (set to zero).
Pedagogical significance
A typical application might set the edges of the basis tetrahedron as unit. The tetrahedron itself may also be defined as the unit of volume (see below).
The four quadrays may be linearly combined to provide integer coordinates for the inverse tetrahedron (0,1,1,1), (1,0,1,1), (1,1,0,1), (1,1,1,0), and for the cube, octahedron, rhombic dodecahedron and cuboctahedron of volumes 3, 4, 6 and 20 respectively, given the starting tetrahedron of unit volume.
For example, given A, B, C, D as (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) respectively, the vertices of an octahedron with the same edge length and volume four would be A + B, A + C, A + D, B + C, B + D, C + D or all eight permutations of {1,1,0,0}.
The 12 permutations of {2,1,1,0} define the vertices of the volume 20 cuboctahedron centered at (0,0,0,0). These vectors point from any given sphere to its 12 surrounding neighbors in the cubic close packing (CCP), equivalently the IVM (isotropic vector matrix) in Synergetics. Therefore CCP ball centers all have non-negative integer coordinates.
Shape Volume Vertex Inventory (sum of Quadrays)
Tetrahedron 1 A,B,C,D
Inverse Tetrahedron 1 E,F,G,H = B+C+D, A+C+D, A+B+D, A+B+C
Duo-Tet Cube 3 A through H
Octahedron 4 I,J,K,L,M,N = A+B, A+C, A+D, B+C, B+D, C+D
Rhombic Dodecahedron 6 A through N
Cuboctahedron 20 O,P,Q,R,S,T = I+J, I+K, I+L, I+M, N+J, N+K; U,V,W,X,Y,Z = N+L, N+M, J+L, L+M, M+K, K+J
If one now calls this volume "4D" as in "four-dimensional" or "four-directional" we have primed the pump for an understanding of R. Buckminster Fuller's "4D geometry," or Synergetics.
In this American transcendentalist philosophy, the regular tetrahedron of edges one, as defined by four intertangent uni-radius balls, is taken as unit of volume.
A set of familiar convex polyhedra, termed "the concentric hierarchy" is nested around it, per the above table, such that the cube has volume 3, the octahedron volume 4, rhombic dodecahedron volume 6, and cuboctahedron volume 20.
See also
• Barycentric coordinates (mathematics)
• Caltrop
• Synergetics coordinates
• Synergetics
• Trilinear coordinates
• Tetrahedron
References
1. Urner, Kirby. "Teaching Object-Oriented Programming with Visual FoxPro." FoxPro Advisor (Advisor Media, March, 1999), page 48 ff.
External links
• Ace, Tom. Quadray formulas
• Ace, Tom. 4D-Quadray correspondence
• Urner, Kirby. The Quadray Papers (Math Forum)
• Urner, Kirby. Generating the Face Centered Cubic lattice (FCC) on Github
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Quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices $A$, $B$, $C$ and $D$ is sometimes denoted as $\square ABCD$.[1]
Quadrilateral
Some types of quadrilaterals
Edges and vertices4
Schläfli symbol{4} (for square)
Areavarious methods;
see below
Internal angle (degrees)90° (for square and rectangle)
Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is[1]
$\angle A+\angle B+\angle C+\angle D=360^{\circ }.$
This is a special case of the n-gon interior angle sum formula: S = (n − 2) × 180°.[2]
All non-self-crossing quadrilaterals tile the plane, by repeated rotation around the midpoints of their edges.[3]
Simple quadrilaterals
Any quadrilateral that is not self-intersecting is a simple quadrilateral.
Convex quadrilateral
In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral.
• Irregular quadrilateral (British English) or trapezium (North American English): no sides are parallel. (In British English, this was once called a trapezoid. For more, see Trapezoid § Trapezium vs Trapezoid)
• Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel. Trapezia (UK) and trapezoids (US) include parallelograms.
• Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
• Parallelogram: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
• Rhombus, rhomb:[1] all four sides are of equal length (equilateral). An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths, and some angles are oblique (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree, some define a rhomboid as a parallelogram that is not a rhombus.[4]
• Rectangle: all four angles are right angles (equiangular). An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
• Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
• Oblong: longer than wide, or wider than long (i.e., a rectangle that is not a square).[5]
• Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.
• Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
• Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle.
• Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
• Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
• Harmonic quadrilateral: the products of the lengths of the opposing sides are equal. It is a type of cyclic quadrilateral.
• Bicentric quadrilateral: it is both tangential and cyclic.
• Orthodiagonal quadrilateral: the diagonals cross at right angles.
• Equidiagonal quadrilateral: the diagonals are of equal length.
• Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
• An equilic quadrilateral has two opposite equal sides that when extended, meet at 60°.
• A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length.[6]
• A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.[7]
• A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.[8]
• A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.[9]
Concave quadrilaterals
In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside the quadrilateral.
• A dart (or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but where one interior angle is reflex. See Kite.
Complex quadrilaterals
A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all on the right as the figure is traced out) add up to 720°.[10]
• Crossed trapezoid (US) or trapezium (Commonwealth):[11] a crossed quadrilateral in which one pair of nonadjacent sides is parallel (like a trapezoid)
• Antiparallelogram: a crossed quadrilateral in which each pair of nonadjacent sides have equal lengths (like a parallelogram)
• Crossed rectangle: an antiparallelogram whose sides are two opposite sides and the two diagonals of a rectangle, hence having one pair of parallel opposite sides
• Crossed square: a special case of a crossed rectangle where two of the sides intersect at right angles
Special line segments
The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices.
The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides.[12] They intersect at the "vertex centroid" of the quadrilateral (see § Remarkable points and lines in a convex quadrilateral below).
The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.[13]
Area of a convex quadrilateral
There are various general formulas for the area K of a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD and d = DA.
Trigonometric formulas
The area can be expressed in trigonometric terms as[14]
$K={\frac {pq}{2}}\sin \theta ,$
where the lengths of the diagonals are p and q and the angle between them is θ.[15] In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to $K={\tfrac {pq}{2}}$ since θ is 90°.
The area can be also expressed in terms of bimedians as[16]
$K=mn\sin \varphi ,$
where the lengths of the bimedians are m and n and the angle between them is φ.
Bretschneider's formula[17][14] expresses the area in terms of the sides and two opposite angles:
${\begin{aligned}K&={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{2}}abcd\;[1+\cos(A+C)]}}\\&={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\left[\cos ^{2}\left({\frac {A+C}{2}}\right)\right]}}\end{aligned}}$
where the sides in sequence are a, b, c, d, where s is the semiperimeter, and A and C are two (in fact, any two) opposite angles. This reduces to Brahmagupta's formula for the area of a cyclic quadrilateral—when A + C = 180° .
Another area formula in terms of the sides and angles, with angle C being between sides b and c, and A being between sides a and d, is
$K={\frac {ad}{2}}\sin {A}+{\frac {bc}{2}}\sin {C}.$
In the case of a cyclic quadrilateral, the latter formula becomes $K={\frac {ad+bc}{2}}\sin {A}.$
In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to $K=ab\cdot \sin {A}.$
Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, as long θ is not 90°:[18]
$K={\frac {\left|\tan \theta \right|}{4}}\cdot \left|a^{2}+c^{2}-b^{2}-d^{2}\right|.$
In the case of a parallelogram, the latter formula becomes $K={\frac {\left|\tan \theta \right|}{2}}\cdot \left|a^{2}-b^{2}\right|.$
Another area formula including the sides a, b, c, d is[16]
$K={\frac {\sqrt {((a^{2}+c^{2})-2x^{2})((b^{2}+d^{2})-2x^{2})}}{2}}\sin {\varphi }$
where x is the distance between the midpoints of the diagonals, and φ is the angle between the bimedians.
The last trigonometric area formula including the sides a, b, c, d and the angle α (between a and b) is:[19]
$K={\frac {ab}{2}}\sin {\alpha }+{\frac {\sqrt {4c^{2}d^{2}-(c^{2}+d^{2}-a^{2}-b^{2}+2ab\cdot \cos {\alpha })^{2}}}{4}},$
which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle α), by just changing the first sign + to -.
Non-trigonometric formulas
The following two formulas express the area in terms of the sides a, b, c and d, the semiperimeter s, and the diagonals p, q:
$K={\sqrt {(s-a)(s-b)(s-c)(s-d)-{\tfrac {1}{4}}(ac+bd+pq)(ac+bd-pq)}},$ [20]
$K={\frac {\sqrt {4p^{2}q^{2}-\left(a^{2}+c^{2}-b^{2}-d^{2}\right)^{2}}}{4}}.$ [21]
The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then pq = ac + bd.
The area can also be expressed in terms of the bimedians m, n and the diagonals p, q:
$K={\frac {\sqrt {(m+n+p)(m+n-p)(m+n+q)(m+n-q)}}{2}},$ [22]
$K={\frac {\sqrt {p^{2}q^{2}-(m^{2}-n^{2})^{2}}}{2}}.$ [23]: Thm. 7
In fact, any three of the four values m, n, p, and q suffice for determination of the area, since in any quadrilateral the four values are related by $p^{2}+q^{2}=2(m^{2}+n^{2}).$[24]: p. 126 The corresponding expressions are:[25]
$K={\frac {\sqrt {[(m+n)^{2}-p^{2}]\cdot [p^{2}-(m-n)^{2}]}}{2}},$
if the lengths of two bimedians and one diagonal are given, and[25]
$K={\frac {\sqrt {[(p+q)^{2}-4m^{2}]\cdot [4m^{2}-(p-q)^{2}]}}{4}},$
if the lengths of two diagonals and one bimedian are given.
Vector formulas
The area of a quadrilateral ABCD can be calculated using vectors. Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then
$K={\frac {|\mathbf {AC} \times \mathbf {BD} |}{2}},$
which is half the magnitude of the cross product of vectors AC and BD. In two-dimensional Euclidean space, expressing vector AC as a free vector in Cartesian space equal to (x1,y1) and BD as (x2,y2), this can be rewritten as:
$K={\frac {|x_{1}y_{2}-x_{2}y_{1}|}{2}}.$
Diagonals
Properties of the diagonals in quadrilaterals
In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length.[26] The list applies to the most general cases, and excludes named subsets.
QuadrilateralBisecting diagonalsPerpendicular diagonalsEqual diagonals
Trapezoid NoSee note 1No
Isosceles trapezoid NoSee note 1Yes
Parallelogram YesNoNo
Kite See note 2YesSee note 2
Rectangle YesNoYes
Rhombus YesYesNo
Square YesYesYes
Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles trapezoids that do have perpendicular diagonals and are not any other named quadrilateral.
Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the kites are not any other named quadrilateral).
Lengths of the diagonals
The lengths of the diagonals in a convex quadrilateral ABCD can be calculated using the law of cosines on each triangle formed by one diagonal and two sides of the quadrilateral. Thus
$p={\sqrt {a^{2}+b^{2}-2ab\cos {B}}}={\sqrt {c^{2}+d^{2}-2cd\cos {D}}}$
and
$q={\sqrt {a^{2}+d^{2}-2ad\cos {A}}}={\sqrt {b^{2}+c^{2}-2bc\cos {C}}}.$
Other, more symmetric formulas for the lengths of the diagonals, are[27]
$p={\sqrt {\frac {(ac+bd)(ad+bc)-2abcd(\cos {B}+\cos {D})}{ab+cd}}}$
and
$q={\sqrt {\frac {(ab+cd)(ac+bd)-2abcd(\cos {A}+\cos {C})}{ad+bc}}}.$
Generalizations of the parallelogram law and Ptolemy's theorem
In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals plus four times the square of the line segment connecting the midpoints of the diagonals. Thus
$a^{2}+b^{2}+c^{2}+d^{2}=p^{2}+q^{2}+4x^{2}$
where x is the distance between the midpoints of the diagonals.[24]: p.126 This is sometimes known as Euler's quadrilateral theorem and is a generalization of the parallelogram law.
The German mathematician Carl Anton Bretschneider derived in 1842 the following generalization of Ptolemy's theorem, regarding the product of the diagonals in a convex quadrilateral[28]
$p^{2}q^{2}=a^{2}c^{2}+b^{2}d^{2}-2abcd\cos {(A+C)}.$
This relation can be considered to be a law of cosines for a quadrilateral. In a cyclic quadrilateral, where A + C = 180°, it reduces to pq = ac + bd. Since cos (A + C) ≥ −1, it also gives a proof of Ptolemy's inequality.
Other metric relations
If X and Y are the feet of the normals from B and D to the diagonal AC = p in a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, then[29]: p.14
$XY={\frac {|a^{2}+c^{2}-b^{2}-d^{2}|}{2p}}.$
In a convex quadrilateral ABCD with sides a = AB, b = BC, c = CD, d = DA, and where the diagonals intersect at E,
$efgh(a+c+b+d)(a+c-b-d)=(agh+cef+beh+dfg)(agh+cef-beh-dfg)$
where e = AE, f = BE, g = CE, and h = DE.[30]
The shape and size of a convex quadrilateral are fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals p, q and the four side lengths a, b, c, d of a quadrilateral are related[14] by the Cayley-Menger determinant, as follows:
$\det {\begin{bmatrix}0&a^{2}&p^{2}&d^{2}&1\\a^{2}&0&b^{2}&q^{2}&1\\p^{2}&b^{2}&0&c^{2}&1\\d^{2}&q^{2}&c^{2}&0&1\\1&1&1&1&0\end{bmatrix}}=0.$
Angle bisectors
The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral[24]: p.127 (that is, the four intersection points of adjacent angle bisectors are concyclic) or they are concurrent. In the latter case the quadrilateral is a tangential quadrilateral.
In quadrilateral ABCD, if the angle bisectors of A and C meet on diagonal BD, then the angle bisectors of B and D meet on diagonal AC.[31]
Bimedians
See also: Varignon's theorem
The bimedians of a quadrilateral are the line segments connecting the midpoints of the opposite sides. The intersection of the bimedians is the centroid of the vertices of the quadrilateral.[14]
The midpoints of the sides of any quadrilateral (convex, concave or crossed) are the vertices of a parallelogram called the Varignon parallelogram. It has the following properties:
• Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
• A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.
• The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.[32]
• The perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.
• The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.
The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are concurrent and are all bisected by their point of intersection.[24]: p.125
In a convex quadrilateral with sides a, b, c and d, the length of the bimedian that connects the midpoints of the sides a and c is
$m={\tfrac {1}{2}}{\sqrt {-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}}$
where p and q are the length of the diagonals.[33] The length of the bimedian that connects the midpoints of the sides b and d is
$n={\tfrac {1}{2}}{\sqrt {a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}}.$
Hence[24]: p.126
$\displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).$
This is also a corollary to the parallelogram law applied in the Varignon parallelogram.
The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence[23]
$m={\tfrac {1}{2}}{\sqrt {2(b^{2}+d^{2})-4x^{2}}}$
and
$n={\tfrac {1}{2}}{\sqrt {2(a^{2}+c^{2})-4x^{2}}}.$
Note that the two opposite sides in these formulas are not the two that the bimedian connects.
In a convex quadrilateral, there is the following dual connection between the bimedians and the diagonals:[29]
• The two bimedians have equal length if and only if the two diagonals are perpendicular.
• The two bimedians are perpendicular if and only if the two diagonals have equal length.
Trigonometric identities
The four angles of a simple quadrilateral ABCD satisfy the following identities:[34]
$\sin {A}+\sin {B}+\sin {C}+\sin {D}=4\sin {\frac {A+B}{2}}\sin {\frac {A+C}{2}}\sin {\frac {A+D}{2}}$
and
${\frac {\tan {A}\tan {B}-\tan {C}\tan {D}}{\tan {A}\tan {C}-\tan {B}\tan {D}}}={\frac {\tan {(A+C)}}{\tan {(A+B)}}}.$
Also,[35]
${\frac {\tan {A}+\tan {B}+\tan {C}+\tan {D}}{\cot {A}+\cot {B}+\cot {C}+\cot {D}}}=\tan {A}\tan {B}\tan {C}\tan {D}.$
In the last two formulas, no angle is allowed to be a right angle, since tan 90° is not defined.
Let $a$, $b$, $c$, $d$ be the sides of a convex quadrilateral, $s$ is the semiperimeter, and $A$ and $C$ are opposite angles, then[36]
$ad\sin ^{2}{\frac {A}{2}}+bc\cos ^{2}{\frac {C}{2}}=(s-a)(s-d)$
and
$bc\sin ^{2}{\frac {C}{2}}+ad\cos ^{2}{\frac {A}{2}}=(s-b)(s-c)$.
We can use these identities to derive the Bretschneider's Formula.
Inequalities
Area
If a convex quadrilateral has the consecutive sides a, b, c, d and the diagonals p, q, then its area K satisfies[37]
$K\leq {\tfrac {1}{4}}(a+c)(b+d)$ with equality only for a rectangle.
$K\leq {\tfrac {1}{4}}(a^{2}+b^{2}+c^{2}+d^{2})$ with equality only for a square.
$K\leq {\tfrac {1}{4}}(p^{2}+q^{2})$ with equality only if the diagonals are perpendicular and equal.
$K\leq {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}$ with equality only for a rectangle.[16]
From Bretschneider's formula it directly follows that the area of a quadrilateral satisfies
$K\leq {\sqrt {(s-a)(s-b)(s-c)(s-d)}}$
with equality if and only if the quadrilateral is cyclic or degenerate such that one side is equal to the sum of the other three (it has collapsed into a line segment, so the area is zero).
The area of any quadrilateral also satisfies the inequality[38]
$\displaystyle K\leq {\tfrac {1}{2}}{\sqrt[{3}]{(ab+cd)(ac+bd)(ad+bc)}}.$
Denoting the perimeter as L, we have[38]: p.114
$K\leq {\tfrac {1}{16}}L^{2},$
with equality only in the case of a square.
The area of a convex quadrilateral also satisfies
$K\leq {\tfrac {1}{2}}pq$
for diagonal lengths p and q, with equality if and only if the diagonals are perpendicular.
Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K and diagonals AC = p, BD = q. Then[39]
$K\leq {\frac {a^{2}+b^{2}+c^{2}+d^{2}+p^{2}+q^{2}+pq-ac-bd}{8}}$ with equality only for a square.
Let a, b, c, d be the lengths of the sides of a convex quadrilateral ABCD with the area K, then the following inequality holds:[40]
$K\leq {\frac {1}{3+{\sqrt {3}}}}(ab+ac+ad+bc+bd+cd)-{\frac {1}{2(1+{\sqrt {3}})^{2}}}(a^{2}+b^{2}+c^{2}+d^{2})$ with equality only for a square.
Diagonals and bimedians
A corollary to Euler's quadrilateral theorem is the inequality
$a^{2}+b^{2}+c^{2}+d^{2}\geq p^{2}+q^{2}$
where equality holds if and only if the quadrilateral is a parallelogram.
Euler also generalized Ptolemy's theorem, which is an equality in a cyclic quadrilateral, into an inequality for a convex quadrilateral. It states that
$pq\leq ac+bd$
where there is equality if and only if the quadrilateral is cyclic.[24]: p.128–129 This is often called Ptolemy's inequality.
In any convex quadrilateral the bimedians m, n and the diagonals p, q are related by the inequality
$pq\leq m^{2}+n^{2},$
with equality holding if and only if the diagonals are equal.[41]: Prop.1 This follows directly from the quadrilateral identity $m^{2}+n^{2}={\tfrac {1}{2}}(p^{2}+q^{2}).$
Sides
The sides a, b, c, and d of any quadrilateral satisfy[42]: p.228, #275
$a^{2}+b^{2}+c^{2}>{\frac {d^{2}}{3}}$
and[42]: p.234, #466
$a^{4}+b^{4}+c^{4}\geq {\frac {d^{4}}{27}}.$
Maximum and minimum properties
Among all quadrilaterals with a given perimeter, the one with the largest area is the square. This is called the isoperimetric theorem for quadrilaterals. It is a direct consequence of the area inequality[38]: p.114
$K\leq {\tfrac {1}{16}}L^{2}$
where K is the area of a convex quadrilateral with perimeter L. Equality holds if and only if the quadrilateral is a square. The dual theorem states that of all quadrilaterals with a given area, the square has the shortest perimeter.
The quadrilateral with given side lengths that has the maximum area is the cyclic quadrilateral.[43]
Of all convex quadrilaterals with given diagonals, the orthodiagonal quadrilateral has the largest area.[38]: p.119 This is a direct consequence of the fact that the area of a convex quadrilateral satisfies
$K={\tfrac {1}{2}}pq\sin {\theta }\leq {\tfrac {1}{2}}pq,$
where θ is the angle between the diagonals p and q. Equality holds if and only if θ = 90°.
If P is an interior point in a convex quadrilateral ABCD, then
$AP+BP+CP+DP\geq AC+BD.$
From this inequality it follows that the point inside a quadrilateral that minimizes the sum of distances to the vertices is the intersection of the diagonals. Hence that point is the Fermat point of a convex quadrilateral.[44]: p.120
Remarkable points and lines in a convex quadrilateral
The centre of a quadrilateral can be defined in several different ways. The "vertex centroid" comes from considering the quadrilateral as being empty but having equal masses at its vertices. The "side centroid" comes from considering the sides to have constant mass per unit length. The usual centre, called just centroid (centre of area) comes from considering the surface of the quadrilateral as having constant density. These three points are in general not all the same point.[45]
The "vertex centroid" is the intersection of the two bimedians.[46] As with any polygon, the x and y coordinates of the vertex centroid are the arithmetic means of the x and y coordinates of the vertices.
The "area centroid" of quadrilateral ABCD can be constructed in the following way. Let Ga, Gb, Gc, Gd be the centroids of triangles BCD, ACD, ABD, ABC respectively. Then the "area centroid" is the intersection of the lines GaGc and GbGd.[47]
In a general convex quadrilateral ABCD, there are no natural analogies to the circumcenter and orthocenter of a triangle. But two such points can be constructed in the following way. Let Oa, Ob, Oc, Od be the circumcenters of triangles BCD, ACD, ABD, ABC respectively; and denote by Ha, Hb, Hc, Hd the orthocenters in the same triangles. Then the intersection of the lines OaOc and ObOd is called the quasicircumcenter, and the intersection of the lines HaHc and HbHd is called the quasiorthocenter of the convex quadrilateral.[47] These points can be used to define an Euler line of a quadrilateral. In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are collinear in this order, and HG = 2GO.[47]
There can also be defined a quasinine-point center E as the intersection of the lines EaEc and EbEd, where Ea, Eb, Ec, Ed are the nine-point centers of triangles BCD, ACD, ABD, ABC respectively. Then E is the midpoint of OH.[47]
Another remarkable line in a convex non-parallelogram quadrilateral is the Newton line, which connects the midpoints of the diagonals, the segment connecting these points being bisected by the vertex centroid. One more interesting line (in some sense dual to the Newton's one) is the line connecting the point of intersection of diagonals with the vertex centroid. The line is remarkable by the fact that it contains the (area) centroid. The vertex centroid divides the segment connecting the intersection of diagonals and the (area) centroid in the ratio 3:1.[48]
For any quadrilateral ABCD with points P and Q the intersections of AD and BC and AB and CD, respectively, the circles (PAB), (PCD), (QAD), and (QBC) pass through a common point M, called a Miquel point.[49]
For a convex quadrilateral ABCD in which E is the point of intersection of the diagonals and F is the point of intersection of the extensions of sides BC and AD, let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. Then there holds: the straight lines NK and ML intersect at point P that is located on the side AB; the straight lines NL and KM intersect at point Q that is located on the side CD. Points P and Q are called "Pascal points" formed by circle ω on sides AB and CD. [50] [51] [52]
Other properties of convex quadrilaterals
• Let exterior squares be drawn on all sides of a quadrilateral. The segments connecting the centers of opposite squares are (a) equal in length, and (b) perpendicular. Thus these centers are the vertices of an orthodiagonal quadrilateral. This is called Van Aubel's theorem.
• For any simple quadrilateral with given edge lengths, there is a cyclic quadrilateral with the same edge lengths.[43]
• The four smaller triangles formed by the diagonals and sides of a convex quadrilateral have the property that the product of the areas of two opposite triangles equals the product of the areas of the other two triangles.[53]
Taxonomy
A hierarchical taxonomy of quadrilaterals is illustrated by the figure to the right. Lower classes are special cases of higher classes they are connected to. Note that "trapezoid" here is referring to the North American definition (the British equivalent is a trapezium). Inclusive definitions are used throughout.
Skew quadrilaterals
See also: Skew polygon
A non-planar quadrilateral is called a skew quadrilateral. Formulas to compute its dihedral angles from the edge lengths and the angle between two adjacent edges were derived for work on the properties of molecules such as cyclobutane that contain a "puckered" ring of four atoms.[54] Historically the term gauche quadrilateral was also used to mean a skew quadrilateral.[55] A skew quadrilateral together with its diagonals form a (possibly non-regular) tetrahedron, and conversely every skew quadrilateral comes from a tetrahedron where a pair of opposite edges is removed.
See also
• Complete quadrangle
• Perpendicular bisector construction of a quadrilateral
• Saccheri quadrilateral
• Types of mesh § Quadrilateral
• Quadrangle (geography)
• Homography - Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography)
References
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3. Martin, George Edward (1982), Transformation geometry, Undergraduate Texts in Mathematics, Springer-Verlag, Theorem 12.1, page 120, doi:10.1007/978-1-4612-5680-9, ISBN 0-387-90636-3, MR 0718119
4. "Archived copy" (PDF). Archived from the original (PDF) on May 14, 2014. Retrieved June 20, 2013.{{cite web}}: CS1 maint: archived copy as title (link)
5. "Rectangles Calculator". Cleavebooks.co.uk. Retrieved 1 March 2022.
6. Keady, G.; Scales, P.; Németh, S. Z. (2004). "Watt Linkages and Quadrilaterals". The Mathematical Gazette. 88 (513): 475–492. doi:10.1017/S0025557200176107. S2CID 125102050.
7. Jobbings, A. K. (1997). "Quadric Quadrilaterals". The Mathematical Gazette. 81 (491): 220–224. doi:10.2307/3619199. JSTOR 3619199. S2CID 250440553.
8. Beauregard, R. A. (2009). "Diametric Quadrilaterals with Two Equal Sides". College Mathematics Journal. 40 (1): 17–21. doi:10.1080/07468342.2009.11922331. S2CID 122206817.
9. Hartshorne, R. (2005). Geometry: Euclid and Beyond. Springer. pp. 429–430. ISBN 978-1-4419-3145-0.
10. "Stars: A Second Look" (PDF). Mysite.mweb.co.za. Archived from the original (PDF) on March 3, 2016. Retrieved March 1, 2022.
11. Butler, David (2016-04-06). "The crossed trapezium". Making Your Own Sense. Retrieved 2017-09-13.
12. E.W. Weisstein. "Bimedian". MathWorld – A Wolfram Web Resource.
13. E.W. Weisstein. "Maltitude". MathWorld – A Wolfram Web Resource.
14. Weisstein, Eric W. "Quadrilateral". mathworld.wolfram.com. Retrieved 2020-09-02.
15. Harries, J. "Area of a quadrilateral," Mathematical Gazette 86, July 2002, 310–311.
16. Josefsson, Martin (2013), "Five Proofs of an Area Characterization of Rectangles" (PDF), Forum Geometricorum, 13: 17–21.
17. R. A. Johnson, Advanced Euclidean Geometry, 2007, Dover Publ., p. 82.
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21. E.W. Weisstein. "Bretschneider's formula". MathWorld – A Wolfram Web Resource.
22. Archibald, R. C., "The Area of a Quadrilateral", American Mathematical Monthly, 29 (1922) pp. 29–36.
23. Josefsson, Martin (2011), "The Area of a Bicentric Quadrilateral" (PDF), Forum Geometricorum, 11: 155–164.
24. Altshiller-Court, Nathan, College Geometry, Dover Publ., 2007.
25. Josefsson, Martin (2016) ‘100.31 Heron-like formulas for quadrilaterals’, The Mathematical Gazette, 100 (549), pp. 505–508.
26. "Diagonals of Quadrilaterals -- Perpendicular, Bisecting or Both". Math.okstate.edu. Retrieved 1 March 2022.
27. Rashid, M. A. & Ajibade, A. O., "Two conditions for a quadrilateral to be cyclic expressed in terms of the lengths of its sides", Int. J. Math. Educ. Sci. Technol., vol. 34 (2003) no. 5, pp. 739–799.
28. Andreescu, Titu & Andrica, Dorian, Complex Numbers from A to...Z, Birkhäuser, 2006, pp. 207–209.
29. Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals" (PDF), Forum Geometricorum, 12: 13–25.
30. Hoehn, Larry (2011), "A New Formula Concerning the Diagonals and Sides of a Quadrilateral" (PDF), Forum Geometricorum, 11: 211–212.
31. Leversha, Gerry, "A property of the diagonals of a cyclic quadrilateral", Mathematical Gazette 93, March 2009, 116–118.
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38. Alsina, Claudi; Nelsen, Roger (2009), When Less is More: Visualizing Basic Inequalities, Mathematical Association of America, p. 68.
39. Dao Thanh Oai, Leonard Giugiuc, Problem 12033, American Mathematical Monthly, March 2018, p. 277
40. Leonard Mihai Giugiuc; Dao Thanh Oai; Kadir Altintas (2018). "An inequality related to the lengths and area of a convex quadrilateral" (PDF). International Journal of Geometry. 7: 81–86.
41. Josefsson, Martin (2014). "Properties of equidiagonal quadrilaterals". Forum Geometricorum. 14: 129–144.
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43. Peter, Thomas, "Maximizing the Area of a Quadrilateral", The College Mathematics Journal, Vol. 34, No. 4 (September 2003), pp. 315–316.
44. Alsina, Claudi; Nelsen, Roger (2010). Charming Proofs : A Journey Into Elegant Mathematics. Mathematical Association of America. pp. 114, 119, 120, 261. ISBN 978-0-88385-348-1.
45. "Two Centers of Mass of a Quadrilateral". Sites.math.washington.edu. Retrieved 1 March 2022.
46. Honsberger, Ross, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Math. Assoc. Amer., 1995, pp. 35–41.
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48. John Boris Miller. "Centroid of a quadrilateral" (PDF). Austmd.org.au. Retrieved March 1, 2022.
49. Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Washington, D.C.: Mathematical Association of America. p. 198. ISBN 9780883858394.
50. David, Fraivert (2019), "Pascal-points quadrilaterals inscribed in a cyclic quadrilateral", The Mathematical Gazette, 103 (557): 233–239, doi:10.1017/mag.2019.54, S2CID 233360695.
51. David, Fraivert (2019), "A Set of Rectangles Inscribed in an Orthodiagonal Quadrilateral and Defined by Pascal-Points Circles", Journal for Geometry and Graphics, 23: 5–27.
52. David, Fraivert (2017), "Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals" (PDF), Forum Geometricorum, 17: 509–526.
53. Josefsson, Martin (2013). "Characterizations of Trapezoids" (PDF). Forum Geometricorum. 13: 23–35.
54. Barnett, M. P.; Capitani, J. F. (2006). "Modular chemical geometry and symbolic calculation". International Journal of Quantum Chemistry. 106 (1): 215–227. Bibcode:2006IJQC..106..215B. doi:10.1002/qua.20807.
55. Hamilton, William Rowan (1850). "On Some Results Obtained by the Quaternion Analysis Respecting the Inscription of "Gauche" Polygons in Surfaces of the Second Order" (PDF). Proceedings of the Royal Irish Academy. 4: 380–387.
External links
Wikimedia Commons has media related to Tetragons.
• "Quadrangle, complete", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Quadrilaterals Formed by Perpendicular Bisectors, Projective Collinearity and Interactive Classification of Quadrilaterals from cut-the-knot
• Definitions and examples of quadrilaterals and Definition and properties of tetragons from Mathopenref
• A (dynamic) Hierarchical Quadrilateral Tree at Dynamic Geometry Sketches
• An extended classification of quadrilaterals Archived 2019-12-30 at the Wayback Machine at Dynamic Math Learning Homepage Archived 2018-08-25 at the Wayback Machine
• The role and function of a hierarchical classification of quadrilaterals by Michael de Villiers
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
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Wikipedia
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Quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.
For the computing company, see Quadrics (company). For quadrics in algebraic geometry, see Quadric (algebraic geometry).
In coordinates x1, x2, ..., xD+1, the general quadric is thus defined by the algebraic equation[1]
$\sum _{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{D+1}P_{i}x_{i}+R=0$
which may be compactly written in vector and matrix notation as:
$xQx^{\mathrm {T} }+Px^{\mathrm {T} }+R=0\,$
where x = (x1, x2, ..., xD+1) is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1) × (D + 1) matrix and P is a (D + 1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any field.
A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see § Normal form of projective quadrics, below.
Euclidean plane
Main article: conic section
As the dimension of a Euclidean plane is two, quadrics in a Euclidean plane have dimension one and are thus plane curves. They are called conic sections, or conics.
Euclidean space
In three-dimensional Euclidean space, quadrics have dimension two, and are known as quadric surfaces. Their quadratic equations have the form
$Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0,$
where $A,B,\ldots ,J$ are real numbers, and at least one of A, B, and C is nonzero.
The quadric surfaces are classified and named by their shape, which corresponds to the orbits under affine transformations. That is, if an affine transformation maps a quadric onto another one, they belong to the same class, and share the same name and many properties.
The principal axis theorem shows that for any (possibly reducible) quadric, a suitable change of Cartesian coordinates or, equivalently, a Euclidean transformation allows putting the equation of the quadric into a unique simple form on which the class of the quadric is immediately visible. This form is called the normal form of the equation, since two quadrics have the same normal form if and only if there is a Euclidean transformation that maps one quadric to the other. The normal forms are as follows:
${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+\varepsilon _{1}{z^{2} \over c^{2}}+\varepsilon _{2}=0,$
${x^{2} \over a^{2}}-{y^{2} \over b^{2}}+\varepsilon _{3}=0$
${x^{2} \over a^{2}}+\varepsilon _{4}=0,$
$z={x^{2} \over a^{2}}+\varepsilon _{5}{y^{2} \over b^{2}},$
where the $\varepsilon _{i}$ are either 1, –1 or 0, except $\varepsilon _{3}$ which takes only the value 0 or 1.
Each of these 17 normal forms[2] corresponds to a single orbit under affine transformations. In three cases there are no real points: $\varepsilon _{1}=\varepsilon _{2}=1$ (imaginary ellipsoid), $\varepsilon _{1}=0,\varepsilon _{2}=1$ (imaginary elliptic cylinder), and $\varepsilon _{4}=1$ (pair of complex conjugate parallel planes, a reducible quadric). In one case, the imaginary cone, there is a single point ($\varepsilon _{1}=1,\varepsilon _{2}=0$). If $\varepsilon _{1}=\varepsilon _{2}=0,$ one has a line (in fact two complex conjugate intersecting planes). For $\varepsilon _{3}=0,$ one has two intersecting planes (reducible quadric). For $\varepsilon _{4}=0,$ one has a double plane. For $\varepsilon _{4}=-1,$ one has two parallel planes (reducible quadric).
Thus, among the 17 normal forms, there are nine true quadrics: a cone, three cylinders (often called degenerate quadrics) and five non-degenerate quadrics (ellipsoid, paraboloids and hyperboloids), which are detailed in the following tables. The eight remaining quadrics are the imaginary ellipsoid (no real point), the imaginary cylinder (no real point), the imaginary cone (a single real point), and the reducible quadrics, which are decomposed in two planes; there are five such decomposed quadrics, depending whether the planes are distinct or not, parallel or not, real or complex conjugate.
Non-degenerate real quadric surfaces
Ellipsoid ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1\,$
Elliptic paraboloid ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-z=0\,$
Hyperbolic paraboloid ${x^{2} \over a^{2}}-{y^{2} \over b^{2}}-z=0\,$
Hyperboloid of one sheet
or
Hyperbolic hyperboloid
${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1\,$
Hyperboloid of two sheets
or
Elliptic hyperboloid
${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1\,$
Degenerate real quadric surfaces
Elliptic cone
or
Conical quadric
${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0\,$
Elliptic cylinder ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\,$
Hyperbolic cylinder ${x^{2} \over a^{2}}-{y^{2} \over b^{2}}=1\,$
Parabolic cylinder $x^{2}+2ay=0\,$
When two or more of the parameters of the canonical equation are equal, one obtains a quadric of revolution, which remains invariant when rotated around an axis (or infinitely many axes, in the case of the sphere).
Quadrics of revolution
Oblate and prolate spheroids (special cases of ellipsoid) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}+{z^{2} \over b^{2}}=1\,$
Sphere (special case of spheroid) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}+{z^{2} \over a^{2}}=1\,$
Circular paraboloid (special case of elliptic paraboloid) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-z=0\,$
Hyperboloid of revolution of one sheet (special case of hyperboloid of one sheet) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=1\,$
Hyperboloid of revolution of two sheets (special case of hyperboloid of two sheets) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=-1\,$
Circular cone (special case of elliptic cone) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=0\,$
Circular cylinder (special case of elliptic cylinder) ${x^{2} \over a^{2}}+{y^{2} \over a^{2}}=1\,$
Definition and basic properties
An affine quadric is the set of zeros of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have real coefficients, and the zeros are points in a Euclidean space. However, most properties remain true when the coefficients belong to any field and the points belong in an affine space. As usual in algebraic geometry, it is often useful to consider points over an algebraically closed field containing the polynomial coefficients, generally the complex numbers, when the coefficients are real.
Many properties becomes easier to state (and to prove) by extending the quadric to the projective space by projective completion, consisting of adding points at infinity. Technically, if
$p(x_{1},\ldots ,x_{n})$
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing p into
$P(X_{0},\ldots ,X_{n})=X_{0}^{2}\,p\left({\frac {X_{1}}{X_{0}}},\ldots ,{\frac {X_{n}}{X_{0}}}\right)$
(this is a polynomial, because the degree of p is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of P.
So, a projective quadric is the set of zeros in a projective space of a homogeneous polynomial of degree two.
As the above process of homogenization can be reverted by setting X0 = 1:
$p(x_{1},\ldots ,x_{n})=P(1,x_{1},\ldots ,x_{n})\,,$
it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the affine equation or the projective equation of a quadric. However, this is not a perfect equivalence; it is generally the case that $P(\mathbf {X} )=0$ will include points with $X_{0}=0$, which are not also solutions of $p(\mathbf {x} )=0$ because these points in projective space correspond to points "at infinity" in affine space.
Equation
A quadric in an affine space of dimension n is the set of zeros of a polynomial of degree 2. That is, it is the set of the points whose coordinates satisfy an equation
$p(x_{1},\ldots ,x_{n})=0,$
where the polynomial p has the form
$p(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}x_{i}x_{j}+\sum _{i=1}^{n}(a_{i,0}+a_{0,i})x_{i}+a_{0,0}\,,$
for a matrix $A=(a_{i,j})$ with $i$ and $j$ running from 0 to $n$. When the characteristic of the field of the coefficients is not two, generally $a_{i,j}=a_{j,i}$ is assumed; equivalently $A=A^{\mathsf {T}}$. When the characteristic of the field of the coefficients is two, generally $a_{i,j}=0$ is assumed when $j<i$; equivalently $A$ is upper triangular.
The equation may be shortened, as the matrix equation
$\mathbf {x} ^{\mathsf {T}}A\mathbf {x} =0\,,$
with
$\mathbf {x} ={\begin{pmatrix}1&x_{1}&\cdots &x_{n}\end{pmatrix}}^{\mathsf {T}}\,.$
The equation of the projective completion is almost identical:
$\mathbf {X} ^{\mathsf {T}}A\mathbf {X} =0,$
with
$\mathbf {X} ={\begin{pmatrix}X_{0}&X_{1}&\cdots &X_{n}\end{pmatrix}}^{\mathsf {T}}.$
These equations define a quadric as an algebraic hypersurface of dimension n – 1 and degree two in a space of dimension n.
A quadric is said to be non-degenerate if the matrix $A$ is invertible.
A non-degenerate quadric is non-singular in the sense that its projective completion has no singular point (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).
The singular points of a degenerate quadric are the points whose projective coordinates belong to the null space of the matrix A.
A quadric is reducible if and only if the rank of A is one (case of a double hyperplane) or two (case of two hyperplanes).
Normal form of projective quadrics
In real projective space, by Sylvester's law of inertia, a non-singular quadratic form P(X) may be put into the normal form
$P(X)=\pm X_{0}^{2}\pm X_{1}^{2}\pm \cdots \pm X_{D+1}^{2}$
by means of a suitable projective transformation (normal forms for singular quadrics can have zeros as well as ±1 as coefficients). For two-dimensional surfaces (dimension D = 2) in three-dimensional space, there are exactly three non-degenerate cases:
$P(X)={\begin{cases}X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\\X_{0}^{2}+X_{1}^{2}+X_{2}^{2}-X_{3}^{2}\\X_{0}^{2}+X_{1}^{2}-X_{2}^{2}-X_{3}^{2}\end{cases}}$
The first case is the empty set.
The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively. These all have positive Gaussian curvature.
The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly ruled surfaces of negative Gaussian curvature.
The degenerate form
$X_{0}^{2}-X_{1}^{2}-X_{2}^{2}=0.\,$
generates the elliptic cylinder, the parabolic cylinder, the hyperbolic cylinder, or the cone, depending on whether the plane at infinity cuts it in a point, a line, two lines, or a nondegenerate conic respectively. These are singly ruled surfaces of zero Gaussian curvature.
We see that projective transformations don't mix Gaussian curvatures of different sign. This is true for general surfaces.[3]
In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.
Rational parametrization
Given a non-singular point A of a quadric, a line passing through A is either tangent to the quadric, or intersects the quadric in exactly one other point (as usual, a line contained in the quadric is considered as a tangent, since it is contained in the tangent hyperplane). This means that the lines passing through A and not tangent to the quadric are in one to one correspondence with the points of the quadric that do not belong to the tangent hyperplane at A. Expressing the points of the quadric in terms of the direction of the corresponding line provides parametric equations of the following forms.
In the case of conic sections (quadric curves), this pametrization establishes a bijection between a projective conic section and a projective line; this bijection is an isomorphism of algebraic curves. In higher dimensions, the parametrization defines a birational map, which is a bijection between dense open subsets of the quadric and a projective space of the same dimension (the topology that is considered is the usual one in the case of a real or complex quadric, or the Zariski topology in all cases). The points of the quadric that are not in the image of this bijection are the points of intersection of the quadric and its tangent hyperplane at A.
In the affine case, the parametrization is a rational parametrization of the form
$x_{i}={\frac {f_{i}(t_{1},\ldots ,t_{n-1})}{f_{0}(t_{1},\ldots ,t_{n-1})}}\quad {\text{for }}i=1,\ldots ,n,$
where $x_{1},\ldots ,x_{n}$ are the coordinates of a point of the quadric, $t_{1},\ldots ,t_{n-1}$ are parameters, and $f_{0},f_{1},\ldots ,f_{n}$ are polynomials of degree at most two.
In the projective case, the parametrization has the form
$X_{i}=F_{i}(T_{1},\ldots ,T_{n})\quad {\text{for }}i=0,\ldots ,n,$
where $X_{0},\ldots ,X_{n}$ are the projective coordinates of a point of the quadric, $T_{1},\ldots ,T_{n}$ are parameters, and $F_{0},\ldots ,F_{n}$ are homogeneous polynomials of degree two.
One passes from one parametrization to the other by putting $x_{i}=X_{i}/X_{0},$ and $t_{i}=T_{i}/T_{n}\,:$
$F_{i}(T_{1},\ldots ,T_{n})=T_{n}^{2}\,f_{i}\!{\left({\frac {T_{1}}{T_{n}}},\ldots ,{\frac {T_{n-1}}{T_{n}}}\right)}.$
For computing the parametrization and proving that the degrees are as asserted, one may proceed as follows in the affine case. One can proceed similarly in the projective case.
Let q be the quadratic polynomial that defines the quadric, and $\mathbf {a} =(a_{1},\ldots a_{n})$ be the coordinate vector of the given point of the quadric (so, $q(\mathbf {a} )=0).$ Let $\mathbf {x} =(x_{1},\ldots x_{n})$ be the coordinate vector of the point of the quadric to be parametrized, and $\mathbf {t} =(t_{1},\ldots ,t_{n-1},1)$ be a vector defining the direction used for the parametrization (directions whose last coordinate is zero are not taken into account here; this means that some points of the affine quadric are not parametrized; one says often that they are parametrized by points at infinity in the space of parameters) . The points of the intersection of the quadric and the line of direction $\mathbf {t} $ passing through $\mathbf {a} $ are the points $\mathbf {x} =\mathbf {a} +\lambda \mathbf {t} $ such that
$q(\mathbf {a} +\lambda \mathbf {t} )=0$
for some value of the scalar $\lambda .$ This is an equation of degree two in $\lambda ,$ except for the values of $\mathbf {t} $ such that the line is tangent to the quadric (in this case, the degree is one if the line is not included in the quadric, or the equation becomes $0=0$ otherwise). The coefficients of $\lambda $ and $\lambda ^{2}$ are respectively of degree at most one and two in $\mathbf {t} .$ As the constant coefficient is $q(\mathbf {a} )=0,$ the equation becomes linear by dividing by $\lambda ,$ and its unique solution is the quotient of a polynomial of degree at most one by a polynomial of degree at most two. Substituting this solution into the expression of $\mathbf {x} ,$ one obtains the desired parametrization as fractions of polynomials of degree at most two.
Example: circle and spheres
Let consider the quadric of equation
$x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}-1=0.$
For $n=2,$ this is the unit circle; for $n=3$ this is the unit sphere; in higher dimension, this is the unit hypersphere.
The point $\mathbf {a} =(0,\ldots ,0,-1)$ belongs to the quadric (the choice of this point among other similar points is only a question of convenience). So, the equation $q(\mathbf {a} +\lambda \mathbf {t} )=0$ of the preceding section becomes
$(\lambda t_{1}^{2})+\cdots +(\lambda t_{n-1})^{2}+(1-\lambda )^{2}-1=0.$
By expanding the squares, simplifying the constant terms, dividing by $\lambda ,$ and solving in $\lambda ,$ one obtains
$\lambda ={\frac {2}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}.$
Substituting this into $\mathbf {x} =\mathbf {a} +\lambda \mathbf {t} $ and simplifying the expression of the last coordinate, one obtains the parametric equation
${\begin{cases}x_{1}={\frac {2t_{1}}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}\\\vdots \\x_{n-1}={\frac {2t_{n-1}}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}\\x_{n}={\frac {1-t_{1}^{2}-\cdots -t_{n-1}^{2}}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}.\end{cases}}$
By homogenizing, one obtains the projective parametrization
${\begin{cases}X_{0}=T_{1}^{2}+\cdots +T_{n}^{2}\\X_{1}=2T_{1}T_{n}\\\vdots \\X_{n-1}=2T_{n-1}T_{n}\\X_{n}=T_{n}^{2}-T_{1}^{2}-\cdots -T_{n-1}^{2}.\end{cases}}$
A straightforward verification shows that this induces a bijection between the points of the quadric such that $X_{n}\neq -X_{0}$ and the points such that $T_{n}\neq 0$ in the projective space of the parameters. On the other hand, all values of $(T_{1},\ldots ,T_{n})$ such that $T_{n}=0$ and $T_{1}^{2}+\cdots +T_{n-1}^{2}\neq 0$ give the point $A.$
In the case of conic sections ($n=2$), there is exactly one point with $T_{n}=0.$ and one has a bijection between the circle and the projective line.
For $n>2,$ there are many points with $T_{n}=0,$ and thus many parameter values for the point $A.$ On the other hand, the other points of the quadric for which $X_{n}=-X_{0}$ (and thus $x_{n}=-1$) cannot be obtained for any value of the parameters. These points are the points of the intersection of the quadric and its tangent plane at $A.$ In this specific case, these points have nonreal complex coordinates, but it suffices to change one sign in the equation of the quadric for producing real points that are not obtained with the resulting parametrization.
Rational points
A quadric is defined over a field $F$ if the coefficients of its equation belong to $F.$ When $F$ is the field $\mathbb {Q} $ of the rational numbers, one can suppose that the coefficients are integers by clearing denominators.
A point of a quadric defined over a field $F$ is said rational over $F$ if its coordinates belong to $F.$ A rational point over the field $\mathbb {R} $ of the real numbers, is called a real point.
A rational point over $\mathbb {Q} $ is called simply a rational point. By clearing denominators, one can suppose and one supposes generally that the projective coordinates of a rational point (in a quadric defined over $\mathbb {Q} $) are integers. Also, by clearing denominators of the coefficients, one supposes generally that all the coefficients of the equation of the quadric and the polynomials occurring in the parametrization are integers.
Finding the rational points of a projective quadric amounts thus to solve a Diophantine equation.
Given a rational point A over a quadric over a field F, the parametrization described in the preceding section provides rational points when the parameters are in F, and, conversely, every rational point of the quadric can be obtained from parameters in F, if the point is not in the tangent hyperplane at A.
It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if F is infinite), and these points can be algorithmically generated as soon one knows one of them.
As said above, in the case of projective quadrics defined over $\mathbb {Q} ,$ the parametrization takes the form
$X_{i}=F_{i}(T_{1},\ldots ,T_{n})\quad {\text{for }}i=0,\ldots ,n,$
where the $F_{i}$ are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that are setwise coprime integers. If $Q(X_{0},\ldots ,X_{n})=0$ is the equation of the quadric, a solution of this equation is said primitive if its components are setwise coprime integers. The primitive solutions are in one to one correspondence with the rational points of the quadric (up to a change of sign of all components of the solution). The non-primitive integer solutions are obtained by multiplying primitive solutions by arbitrary integers; so they do not deserve a specific study. However, setwise coprime parameters can produce non-primitive solutions, and one may have to divide by a greatest common divisor to arrive at the associated primitive solution.
Pythagorean triples
This is well illustrated by Pythagorean triples. A Pythagorean triple is a triple $(a,b,c)$ of positive integers such that $a^{2}+b^{2}=c^{2}.$ A Pythagorean triple is primitive if $a,b,c$ are setwise coprime, or, equivalently, if any of the three pairs $(a,b),$ $(b,c)$ and $(a,c)$ is coprime.
By choosing $A=(-1,0,1),$ the above method provides the parametrization
${\begin{cases}a=m^{2}-n^{2}\\b=2mn\\c=m^{2}+m^{2}\end{cases}}$
for the quadric of equation $a^{2}+b^{2}-c^{2}=0.$ (The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).
If m and n are coprime integers such that $m>n>0,$ the resulting triple is a Pythagorean triple. If one of m and n is even and the other is odd, this resulting triple is primitive; otherwise, m and n are both odd, and one obtains a primitive triple by dividing by 2.
In summary, the primitive Pythagorean triples with $b$ even are obtained as
$a=m^{2}-n^{2},\quad b=2mn,\quad c=m^{2}+n^{2},$
with m and n coprime integers such that one is even and $m>n>0$ (this is Euclid's formula). The primitive Pythagorean triples with $b$ odd are obtained as
$a={\frac {m^{2}-n^{2}}{2}},\quad b=mn,\quad c={\frac {m^{2}+n^{2}}{2}},$
with m and n coprime odd integers such that $m>n>0.$
As the exchange of a and b transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples.
Projective quadrics over fields
The definition of a projective quadric in a real projective space (see above) can be formally adapted by defining a projective quadric in an n-dimensional projective space over a field. In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space.[4]
Quadratic form
Let $K$ be a field and $V$ a vector space over $K$. A mapping $q$ from $V$ to $K$ such that
(Q1) $\;q(\lambda {\vec {x}})=\lambda ^{2}q({\vec {x}})\;$ for any $\lambda \in K$ and ${\vec {x}}\in V$.
(Q2) $\;f({\vec {x}},{\vec {y}}):=q({\vec {x}}+{\vec {y}})-q({\vec {x}})-q({\vec {y}})\;$ is a bilinear form.
is called quadratic form. The bilinear form $f$ is symmetric.
In case of $\operatorname {char} K\neq 2$ the bilinear form is $f({\vec {x}},{\vec {x}})=2q({\vec {x}})$, i.e. $f$ and $q$ are mutually determined in a unique way.
In case of $\operatorname {char} K=2$ (that means: $1+1=0$) the bilinear form has the property $f({\vec {x}},{\vec {x}})=0$, i.e. $f$ is symplectic.
For $V=K^{n}\ $ and $\ {\vec {x}}=\sum _{i=1}^{n}x_{i}{\vec {e}}_{i}\quad $ ($\{{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}\}$ is a base of $V$) $\ q$ has the familiar form
$q({\vec {x}})=\sum _{1=i\leq k}^{n}a_{ik}x_{i}x_{k}\ {\text{ with }}\ a_{ik}:=f({\vec {e}}_{i},{\vec {e}}_{k})\ {\text{ for }}\ i\neq k\ {\text{ and }}\ a_{ii}:=q({\vec {e}}_{i})\ $ and
$f({\vec {x}},{\vec {y}})=\sum _{1=i\leq k}^{n}a_{ik}(x_{i}y_{k}+x_{k}y_{i})$.
For example:
$n=3,\quad q({\vec {x}})=x_{1}x_{2}-x_{3}^{2},\quad f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}\;.$
n-dimensional projective space over a field
Let $K$ be a field, $2\leq n\in \mathbb {N} $,
$V_{n+1}$ an (n + 1)-dimensional vector space over the field $K,$
$\langle {\vec {x}}\rangle $ the 1-dimensional subspace generated by ${\vec {0}}\neq {\vec {x}}\in V_{n+1}$,
${\mathcal {P}}=\{\langle {\vec {x}}\rangle \mid {\vec {x}}\in V_{n+1}\},\ $ the set of points ,
${\mathcal {G}}=\{{\text{2-dimensional subspaces of }}V_{n+1}\},\ $ the set of lines.
$P_{n}(K)=({\mathcal {P}},{\mathcal {G}})\ $ is the n-dimensional projective space over $K$.
The set of points contained in a $(k+1)$-dimensional subspace of $V_{n+1}$ is a $k$-dimensional subspace of $P_{n}(K)$. A 2-dimensional subspace is a plane.
In case of $\;n>3\;$ a $(n-1)$-dimensional subspace is called hyperplane.
Projective quadric
A quadratic form $q$ on a vector space $V_{n+1}$ defines a quadric ${\mathcal {Q}}$ in the associated projective space ${\mathcal {P}},$ as the set of the points $\langle {\vec {x}}\rangle \in {\mathcal {P}}$ such that $q({\vec {x}})=0$. That is,
${\mathcal {Q}}=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid q({\vec {x}})=0\}.$
Examples in $P_{2}(K)$.:
(E1): For $\;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;$ one obtains a conic.
(E2): For $\;q({\vec {x}})=x_{1}x_{2}\;$ one obtains the pair of lines with the equations $x_{1}=0$ and $x_{2}=0$, respectively. They intersect at point $\langle (0,0,1)^{\text{T}}\rangle $;
For the considerations below it is assumed that ${\mathcal {Q}}\neq \emptyset $.
Polar space
For point $P=\langle {\vec {p}}\rangle \in {\mathcal {P}}$ the set
$P^{\perp }:=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid f({\vec {p}},{\vec {x}})=0\}$
is called polar space of $P$ (with respect to $q$).
If $\;f({\vec {p}},{\vec {x}})=0\;$ for any ${\vec {x}}$, one obtains $P^{\perp }={\mathcal {P}}$.
If $\;f({\vec {p}},{\vec {x}})\neq 0\;$ for at least one ${\vec {x}}$, the equation $\;f({\vec {p}},{\vec {x}})=0\;$is a non trivial linear equation which defines a hyperplane. Hence
$P^{\perp }$ is either a hyperplane or ${\mathcal {P}}$.
Intersection with a line
For the intersection of an arbitrary line $g$ with a quadric ${\mathcal {Q}}$, the following cases may occur:
a) $g\cap {\mathcal {Q}}=\emptyset \;$ and $g$ is called exterior line
b) $g\subset {\mathcal {Q}}\;$ and $g$ is called a line in the quadric
c) $|g\cap {\mathcal {Q}}|=1\;$ and $g$ is called tangent line
d) $|g\cap {\mathcal {Q}}|=2\;$ and $g$ is called secant line.
Proof: Let $g$ be a line, which intersects ${\mathcal {Q}}$ at point $\;U=\langle {\vec {u}}\rangle \;$ and $\;V=\langle {\vec {v}}\rangle \;$ is a second point on $g$. From $\;q({\vec {u}})=0\;$ one obtains
$q(x{\vec {u}}+{\vec {v}})=q(x{\vec {u}})+q({\vec {v}})+f(x{\vec {u}},{\vec {v}})=q({\vec {v}})+xf({\vec {u}},{\vec {v}})\;.$
I) In case of $g\subset U^{\perp }$ the equation $f({\vec {u}},{\vec {v}})=0$ holds and it is $\;q(x{\vec {u}}+{\vec {v}})=q({\vec {v}})\;$ for any $x\in K$. Hence either $\;q(x{\vec {u}}+{\vec {v}})=0\;$ for any $x\in K$ or $\;q(x{\vec {u}}+{\vec {v}})\neq 0\;$ for any $x\in K$, which proves b) and b').
II) In case of $g\not \subset U^{\perp }$ one obtains $f({\vec {u}},{\vec {v}})\neq 0$ and the equation $\;q(x{\vec {u}}+{\vec {v}})=q({\vec {v}})+xf({\vec {u}},{\vec {v}})=0\;$ has exactly one solution $x$. Hence: $|g\cap {\mathcal {Q}}|=2$, which proves c).
Additionally the proof shows:
A line $g$ through a point $P\in {\mathcal {Q}}$ is a tangent line if and only if $g\subset P^{\perp }$.
f-radical, q-radical
In the classical cases $K=\mathbb {R} $ or $\mathbb {C} $ there exists only one radical, because of $\operatorname {char} K\neq 2$ and $f$ and $q$ are closely connected. In case of $\operatorname {char} K=2$ the quadric ${\mathcal {Q}}$ is not determined by $f$ (see above) and so one has to deal with two radicals:
a) ${\mathcal {R}}:=\{P\in {\mathcal {P}}\mid P^{\perp }={\mathcal {P}}\}$ is a projective subspace. ${\mathcal {R}}$ is called f-radical of quadric ${\mathcal {Q}}$.
b) ${\mathcal {S}}:={\mathcal {R}}\cap {\mathcal {Q}}$ is called singular radical or $q$-radical of ${\mathcal {Q}}$.
c) In case of $\operatorname {char} K\neq 2$ one has ${\mathcal {R}}={\mathcal {S}}$.
A quadric is called non-degenerate if ${\mathcal {S}}=\emptyset $.
Examples in $P_{2}(K)$ (see above):
(E1): For $\;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;$ (conic) the bilinear form is $f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}\;.$
In case of $\operatorname {char} K\neq 2$ the polar spaces are never ${\mathcal {P}}$. Hence ${\mathcal {R}}={\mathcal {S}}=\emptyset $.
In case of $\operatorname {char} K=2$ the bilinear form is reduced to $f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}\;$ and ${\mathcal {R}}=\langle (0,0,1)^{\text{T}}\rangle \notin {\mathcal {Q}}$. Hence ${\mathcal {R}}\neq {\mathcal {S}}=\emptyset \;.$ In this case the f-radical is the common point of all tangents, the so called knot.
In both cases $S=\emptyset $ and the quadric (conic) ist non-degenerate.
(E2): For $\;q({\vec {x}})=x_{1}x_{2}\;$ (pair of lines) the bilinear form is $f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}\;$ and ${\mathcal {R}}=\langle (0,0,1)^{\text{T}}\rangle ={\mathcal {S}}\;,$ the intersection point.
In this example the quadric is degenerate.
Symmetries
A quadric is a rather homogeneous object:
For any point $P\notin {\mathcal {Q}}\cup {\mathcal {R}}\;$ there exists an involutorial central collineation $\sigma _{P}$ with center $P$ and $\sigma _{P}({\mathcal {Q}})={\mathcal {Q}}$.
Proof: Due to $P\notin {\mathcal {Q}}\cup {\mathcal {R}}$ the polar space $P^{\perp }$ is a hyperplane.
The linear mapping
$\varphi :{\vec {x}}\rightarrow {\vec {x}}-{\frac {f({\vec {p}},{\vec {x}})}{q({\vec {p}})}}{\vec {p}}$ :{\vec {x}}\rightarrow {\vec {x}}-{\frac {f({\vec {p}},{\vec {x}})}{q({\vec {p}})}}{\vec {p}}}
induces an involutorial central collineation $\sigma _{P}$ with axis $P^{\perp }$ and centre $P$ which leaves ${\mathcal {Q}}$ invariant.
In the case of $\operatorname {char} K\neq 2$, the mapping $\varphi $ produces the familiar shape $\;\varphi :{\vec {x}}\rightarrow {\vec {x}}-2{\frac {f({\vec {p}},{\vec {x}})}{f({\vec {p}},{\vec {p}})}}{\vec {p}}\;$ :{\vec {x}}\rightarrow {\vec {x}}-2{\frac {f({\vec {p}},{\vec {x}})}{f({\vec {p}},{\vec {p}})}}{\vec {p}}\;} with $\;\varphi ({\vec {p}})=-{\vec {p}}$ and $\;\varphi ({\vec {x}})={\vec {x}}\;$ for any $\langle {\vec {x}}\rangle \in P^{\perp }$.
Remark:
a) An exterior line, a tangent line or a secant line is mapped by the involution $\sigma _{P}$ on an exterior, tangent and secant line, respectively.
b) ${\mathcal {R}}$ is pointwise fixed by $\sigma _{P}$.
q-subspaces and index of a quadric
A subspace $\;{\mathcal {U}}\;$ of $P_{n}(K)$ is called $q$-subspace if $\;{\mathcal {U}}\subset {\mathcal {Q}}\;$
For example: points on a sphere or lines on a hyperboloid (s. below).
Any two maximal $q$-subspaces have the same dimension $m$.[5]
Let be $m$ the dimension of the maximal $q$-subspaces of ${\mathcal {Q}}$ then
The integer $\;i:=m+1\;$ is called index of ${\mathcal {Q}}$.
Theorem: (BUEKENHOUT)[6]
For the index $i$ of a non-degenerate quadric ${\mathcal {Q}}$ in $P_{n}(K)$ the following is true:
$i\leq {\frac {n+1}{2}}$.
Let be ${\mathcal {Q}}$ a non-degenerate quadric in $P_{n}(K),n\geq 2$, and $i$ its index.
In case of $i=1$ quadric ${\mathcal {Q}}$ is called sphere (or oval conic if $n=2$).
In case of $i=2$ quadric ${\mathcal {Q}}$ is called hyperboloid (of one sheet).
Examples:
a) Quadric ${\mathcal {Q}}$ in $P_{2}(K)$ with form $\;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;$ is non-degenerate with index 1.
b) If polynomial $\;p(\xi )=\xi ^{2}+a_{0}\xi +b_{0}\;$ is irreducible over $K$ the quadratic form $\;q({\vec {x}})=x_{1}^{2}+a_{0}x_{1}x_{2}+b_{0}x_{2}^{2}-x_{3}x_{4}\;$ gives rise to a non-degenerate quadric ${\mathcal {Q}}$ in $P_{3}(K)$ of index 1 (sphere). For example: $\;p(\xi )=\xi ^{2}+1\;$ is irreducible over $\mathbb {R} $ (but not over $\mathbb {C} $ !).
c) In $P_{3}(K)$ the quadratic form $\;q({\vec {x}})=x_{1}x_{2}+x_{3}x_{4}\;$ generates a hyperboloid.
Generalization of quadrics: quadratic sets
It is not reasonable to formally extend the definition of quadrics to spaces over genuine skew fields (division rings). Because one would obtain secants bearing more than 2 points of the quadric which is totally different from usual quadrics.[7][8][9] The reason is the following statement.
A division ring $K$ is commutative if and only if any equation $x^{2}+ax+b=0,\ a,b\in K$, has at most two solutions.
There are generalizations of quadrics: quadratic sets.[10] A quadratic set is a set of points of a projective space with the same geometric properties as a quadric: every line intersects a quadratic set in at most two points or is contained in the set.
See also
• Klein quadric
• Rotation of axes
• Superquadrics
• Translation of axes
References
1. Silvio Levy Quadrics in "Geometry Formulas and Facts", excerpted from 30th Edition of CRC Standard Mathematical Tables and Formulas, CRC Press, from The Geometry Center at University of Minnesota
2. Stewart Venit and Wayne Bishop, Elementary Linear Algebra (fourth edition), International Thompson Publishing, 1996.
3. S. Lazebnik and J. Ponce, "The Local Projective Shape of Smooth Surfaces and Their Outlines" (PDF)., Proposition 1
4. Beutelspacher/Rosenbaum p.158
5. Beutelpacher/Rosenbaum, p.139
6. F. Buekenhout: Ensembles Quadratiques des Espace Projective, Math. Teitschr. 110 (1969), p. 306-318.
7. R. Artzy: The Conic $y=x^{2}$ in Moufang Planes, Aequat.Mathem. 6 (1971), p. 31-35
8. E. Berz: Kegelschnitte in Desarguesschen Ebenen, Math. Zeitschr. 78 (1962), p. 55-8
9. external link E. Hartmann: Planar Circle Geometries, p. 123
10. Beutelspacher/Rosenbaum: p. 135
Bibliography
• M. Audin: Geometry, Springer, Berlin, 2002, ISBN 978-3-540-43498-6, p. 200.
• M. Berger: Problem Books in Mathematics, ISSN 0941-3502, Springer New York, pp 79–84.
• A. Beutelspacher, U. Rosenbaum: Projektive Geometrie, Vieweg + Teubner, Braunschweig u. a. 1992, ISBN 3-528-07241-5, p. 159.
• P. Dembowski: Finite Geometries, Springer, 1968, ISBN 978-3-540-61786-0, p. 43.
• Iskovskikh, V.A. (2001) [1994], "Quadric", Encyclopedia of Mathematics, EMS Press
• Weisstein, Eric W. "Quadric". MathWorld.
External links
• Interactive Java 3D models of all quadric surfaces
• Lecture Note Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes, p. 117
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Wikipedia
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Quadric (algebraic geometry)
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface
$xy=zw$
This article is about quadrics in algebraic geometry. For quadrics over the real numbers, see quadric.
in projective space ${\mathbf {P} }^{3}$ over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry.
Many properties of quadrics hold more generally for projective homogeneous varieties. Another generalization of quadrics is provided by Fano varieties.
Basic properties
By definition, a quadric X of dimension n over a field k is the subspace of $\mathbf {P} ^{n+1}$ defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables $x_{0},\ldots ,x_{n+1}$. (A homogeneous polynomial is also called a form, and so q may be called a quadratic form.) If q is the product of two linear forms, then X is the union of two hyperplanes. It is common to assume that $n\geq 1$ and q is irreducible, which excludes that special case.
Here algebraic varieties over a field k are considered as a special class of schemes over k. When k is algebraically closed, one can also think of a projective variety in a more elementary way, as a subset of ${\mathbf {P} }^{N}(k)=(k^{N+1}-0)/k^{*}$ defined by homogeneous polynomial equations with coefficients in k.
If q can be written (after some linear change of coordinates) as a polynomial in a proper subset of the variables, then X is the projective cone over a lower-dimensional quadric. It is reasonable to focus attention on the case where X is not a cone. For k of characteristic not 2, X is not a cone if and only if X is smooth over k. When k has characteristic not 2, smoothness of a quadric is also equivalent to the Hessian matrix of q having nonzero determinant, or to the associated bilinear form b(x,y) = q(x+y) – q(x) – q(y) being nondegenerate. In general, for k of characteristic not 2, the rank of a quadric means the rank of the Hessian matrix. A quadric of rank r is an iterated cone over a smooth quadric of dimension r − 2.[1]
It is a fundamental result that a smooth quadric over a field k is rational over k if and only if X has a k-rational point.[2] That is, if there is a solution of the equation q = 0 of the form $(a_{0},\ldots ,a_{n+1})$ with $a_{0},\ldots ,a_{n+1}$ in k, not all zero (hence corresponding to a point in projective space), then there is a one-to-one correspondence defined by rational functions over k between ${\mathbf {P} }^{n}$ minus a lower-dimensional subset and X minus a lower-dimensional subset. For example, if k is infinite, it follows that if X has one k-rational point then it has infinitely many. This equivalence is proved by stereographic projection. In particular, every quadric over an algebraically closed field is rational.
A quadric over a field k is called isotropic if it has a k-rational point. An example of an anisotropic quadric is the quadric
$x_{0}^{2}+x_{1}^{2}+\cdots +x_{n+1}^{2}=0$
in projective space ${\mathbf {P} }^{n+1}$ over the real numbers R.
Linear subspaces of quadrics
A central part of the geometry of quadrics is the study of the linear spaces that they contain. (In the context of projective geometry, a linear subspace of ${\mathbf {P} }^{N}$ is isomorphic to ${\mathbf {P} }^{a}$ for some $a\leq N$.) A key point is that every linear space contained in a smooth quadric has dimension at most half the dimension of the quadric. Moreover, when k is algebraically closed, this is an optimal bound, meaning that every smooth quadric of dimension n over k contains a linear subspace of dimension $\lfloor n/2\rfloor $.[3]
Over any field k, a smooth quadric of dimension n is called split if it contains a linear space of dimension $\lfloor n/2\rfloor $ over k. Thus every smooth quadric over an algebraically closed field is split. If a quadric X over a field k is split, then it can be written (after a linear change of coordinates) as
$x_{0}x_{1}+x_{2}x_{3}+\cdots +x_{2m-2}x_{2m-1}+x_{2m}^{2}=0$
if X has dimension 2m − 1, or
$x_{0}x_{1}+x_{2}x_{3}+\cdots +x_{2m}x_{2m+1}=0$
if X has dimension 2m.[4] In particular, over an algebraically closed field, there is only one smooth quadric of each dimension, up to isomorphism.
For many applications, it is important to describe the space Y of all linear subspaces of maximal dimension in a given smooth quadric X. (For clarity, assume that X is split over k.) A striking phenomenon is that Y is connected if X has odd dimension, whereas it has two connected components if X has even dimension. That is, there are two different "types" of maximal linear spaces in X when X has even dimension. The two families can be described by: for a smooth quadric X of dimension 2m, fix one m-plane Q contained in X. Then the two types of m-planes P contained in X are distinguished by whether the dimension of the intersection $P\cap Q$ is even or odd.[5] (The dimension of the empty set is taken to be −1 here.)
Low-dimensional quadrics
Let X be a split quadric over a field k. (In particular, X can be any smooth quadric over an algebraically closed field.) In low dimensions, X and the linear spaces it contains can be described as follows.
• A quadric curve in $\mathbf {P} ^{2}$ is called a conic. A split conic over k is isomorphic to the projective line $\mathbf {P} ^{1}$ over k, embedded in $\mathbf {P} ^{2}$ by the 2nd Veronese embedding.[6] (For example, ellipses, parabolas and hyperbolas are different kinds of conics in the affine plane over R, but their closures in the projective plane are all isomorphic to $\mathbf {P} ^{1}$ over R.)
• A split quadric surface X is isomorphic to $\mathbf {P} ^{1}\times \mathbf {P} ^{1}$, embedded in $\mathbf {P} ^{3}$ by the Segre embedding. The space of lines in the quadric surface X has two connected components, each isomorphic to $\mathbf {P} ^{1}$.[7]
• A split quadric 3-fold X can be viewed as an isotropic Grassmannian for the symplectic group Sp(4,k). (This is related to the exceptional isomorphism of linear algebraic groups between SO(5,k) and $\operatorname {Sp} (4,k)/\{\pm 1\}$.) Namely, given a 4-dimensional vector space V with a symplectic form, the quadric 3-fold X can be identified with the space LGr(2,4) of 2-planes in V on which the form restricts to zero. Furthermore, the space of lines in the quadric 3-fold X is isomorphic to $\mathbf {P} ^{3}$.[8]
• A split quadric 4-fold X can be viewed as the Grassmannian Gr(2,4), the space of 2-planes in a 4-dimensional vector space (or equivalently, of lines in $\mathbf {P} ^{3}$). (This is related to the exceptional isomorphism of linear algebraic groups between SO(6,k) and $\operatorname {SL} (4,k)/\{\pm 1\}$.) The space of 2-planes in the quadric 4-fold X has two connected components, each isomorphic to $\mathbf {P} ^{3}$.[9]
• The space of 2-planes in a split quadric 5-fold is isomorphic to a split quadric 6-fold. Likewise, both components of the space of 3-planes in a split quadric 6-fold are isomorphic to a split quadric 6-fold. (This is related to the phenomenon of triality for the group Spin(8).)
As these examples suggest, the space of m-planes in a split quadric of dimension 2m always has two connected components, each isomorphic to the isotropic Grassmannian of (m − 1)-planes in a split quadric of dimension 2m − 1.[10] Any reflection in the orthogonal group maps one component isomorphically to the other.
The Bruhat decomposition
A smooth quadric over a field k is a projective homogeneous variety for the orthogonal group (and for the special orthogonal group), viewed as linear algebraic groups over k. Like any projective homogeneous variety for a split reductive group, a split quadric X has an algebraic cell decomposition, known as the Bruhat decomposition. (In particular, this applies to every smooth quadric over an algebraically closed field.) That is, X can be written as a finite union of disjoint subsets that are isomorphic to affine spaces over k of various dimensions. (For projective homogeneous varieties, the cells are called Schubert cells, and their closures are called Schubert varieties.) Cellular varieties are very special among all algebraic varieties. For example, a cellular variety is rational, and (for k = C) the Hodge theory of a smooth projective cellular variety is trivial, in the sense that $h^{p,q}(X)=0$ for $p\neq q$. For a cellular variety, the Chow group of algebraic cycles on X is the free abelian group on the set of cells, as is the integral homology of X (if k = C).[11]
A split quadric X of dimension n has only one cell of each dimension r, except in the middle dimension of an even-dimensional quadric, where there are two cells. The corresponding cell closures (Schubert varieties) are:[12]
• For $0\leq r<n/2$, a linear space $\mathbf {P} ^{r}$ contained in X.
• For r = n/2, both Schubert varieties are linear spaces $\mathbf {P} ^{r}$ contained in X, one from each of the two families of middle-dimensional linear spaces (as described above).
• For $n/2<r\leq n$, the Schubert variety of dimension r is the intersection of X with a linear space of dimension r + 1 in $\mathbf {P} ^{n+1}$; so it is an r-dimensional quadric. It is the iterated cone over a smooth quadric of dimension 2r − n.
Using the Bruhat decomposition, it is straightforward to compute the Chow ring of a split quadric of dimension n over a field, as follows.[13] When the base field is the complex numbers, this is also the integral cohomology ring of a smooth quadric, with $CH^{j}$ mapping isomorphically to $H^{2j}$. (The cohomology in odd degrees is zero.)
• For n = 2m − 1, $CH^{*}(X)\cong \mathbb {Z} [h,l]/(h^{m}-2l,l^{2})$, where |h| = 1 and |l| = m.
• For n = 2m, $CH^{*}(X)\cong \mathbb {Z} [h,l]/(h^{m+1}-2hl,l^{2}-ah^{m}l)$, where |h| = 1 and |l| = m, and a is 0 for m odd and 1 for m even.
Here h is the class of a hyperplane section and l is the class of a maximal linear subspace of X. (For n = 2m, the class of the other type of maximal linear subspace is $h^{m}-l$.) This calculation shows the importance of the linear subspaces of a quadric: the Chow ring of all algebraic cycles on X is generated by the "obvious" element h (pulled back from the class $c_{1}O(1)$ of a hyperplane in ${\mathbf {P} }^{n+1}$) together with the class of a maximal linear subspace of X.
Isotropic Grassmannians and the spinor variety
The space of r-planes in a smooth n-dimensional quadric (like the quadric itself) is a projective homogeneous variety, known as the isotropic Grassmannian or orthogonal Grassmannian OGr(r + 1, n + 2). (The numbering refers to the dimensions of the corresponding vector spaces. In the case of middle-dimensional linear subspaces of an quadric of even dimension 2m, one writes $\operatorname {OGr} _{+}(m+1,2m+2)$ for one of the two connected components.) As a result, the isotropic Grassmannians of a split quadric over a field also have algebraic cell decompositions.
The isotropic Grassmannian W = OGr(m,2m + 1) of (m − 1)-planes in a smooth quadric of dimension 2m − 1 is also called the spinor variety, of dimension m(m + 1)/2. (Another description of the spinor variety is as $\operatorname {OGr} _{+}(m+1,2m+2)$.[10]) To explain the name: the smallest SO(2m + 1)-equivariant projective embedding of W lands in projective space of dimension $2^{m}-1$.[14] The action of SO(2m + 1) on this projective space does not come from a linear representation of SO(2m+1) over k, but rather from a representation of its simply connected double cover, the spin group Spin(2m + 1) over k. This is called the spin representation of Spin(2m + 1), of dimension $2^{m}$.
Over the complex numbers, the isotropic Grassmannian OGr(r + 1, n + 2) of r-planes in an n-dimensional quadric X is a homogeneous space for the complex algebraic group $G=\operatorname {SO} (n+2,\mathbf {C} )$, and also for its maximal compact subgroup, the compact Lie group SO(n + 2). From the latter point of view, this isotropic Grassmannian is
$\operatorname {SO} (n+2)/(\operatorname {U} (r+1)\times \operatorname {SO} (n-2r)),$
where U(r+1) is the unitary group. For r = 0, the isotropic Grassmannian is the quadric itself, which can therefore be viewed as
$\operatorname {SO} (n+2)/(\operatorname {U} (1)\times \operatorname {SO} (n)).$
For example, the complex spinor variety OGr(m, 2m + 1) can be viewed as SO(2m + 1)/U(m), and also as SO(2m+2)/U(m+1). These descriptions can be used to compute the cohomology ring (or equivalently the Chow ring) of the spinor variety:
$CH^{*}\operatorname {OGr} (m,2m+1)\cong \mathbb {Z} [e_{1},\ldots ,e_{m}]/(e_{j}^{2}-2e_{j-1}e_{j+1}+2e_{j-2}e_{j+2}-\cdots +(-1)^{j}e_{2j}=0{\text{ for all }}j),$
where the Chern classes $c_{j}$ of the natural rank-m vector bundle are equal to $2e_{j}$.[15] Here $e_{j}$ is understood to mean 0 for j > m.
Spinor bundles on quadrics
The spinor bundles play a special role among all vector bundles on a quadric, analogous to the maximal linear subspaces among all subvarieties of a quadric. To describe these bundles, let X be a split quadric of dimension n over a field k. The special orthogonal group SO(n+2) over k acts on X, and therefore so does its double cover, the spin group G = Spin(n+2) over k. In these terms, X is a homogeneous space G/P, where P is a maximal parabolic subgroup of G. The semisimple part of P is the spin group Spin(n), and there is a standard way to extend the spin representations of Spin(n) to representations of P. (There are two spin representations $V_{+},V_{-}$ for n = 2m, each of dimension $2^{m-1}$, and one spin representation V for n = 2m − 1, of dimension $2^{m-1}$.) Then the spinor bundles on the quadric X = G/P are defined as the G-equivariant vector bundles associated to these representations of P. So there are two spinor bundles $S_{+},S_{-}$ of rank $2^{m-1}$ for n = 2m, and one spinor bundle S of rank $2^{m-1}$ for n = 2m − 1. For n even, any reflection in the orthogonal group switches the two spinor bundles on X.[14]
For example, the two spinor bundles on a quadric surface $X\cong \mathbf {P} ^{1}\times \mathbf {P} ^{1}$ are the line bundles O(−1,0) and O(0,−1). The spinor bundle on a quadric 3-fold X is the natural rank-2 subbundle on X viewed as the isotropic Grassmannian of 2-planes in a 4-dimensional symplectic vector space.
To indicate the significance of the spinor bundles: Mikhail Kapranov showed that the bounded derived category of coherent sheaves on a split quadric X over a field k has a full exceptional collection involving the spinor bundles, along with the "obvious" line bundles O(j) restricted from projective space:
$D^{b}(X)=\langle S_{+},S_{-},O,O(1),\ldots ,O(n-1)\rangle $
if n is even, and
$D^{b}(X)=\langle S,O,O(1),\ldots ,O(n-1)\rangle $
if n is odd.[16] Concretely, this implies the split case of Richard Swan's calculation of the Grothendieck group of algebraic vector bundles on a smooth quadric; it is the free abelian group
$K_{0}(X)=\mathbb {Z} \{S_{+},S_{-},O,O(1),\ldots ,O(n-1)\}$
for n even, and
$K_{0}(X)=\mathbb {Z} \{S,O,O(1),\ldots ,O(n-1)\}$
for n odd.[17] When k = C, the topological K-group $K^{0}(X)$ (of continuous complex vector bundles on the quadric X) is given by the same formula, and $K^{1}(X)$ is zero.
Notes
1. Harris (1995), Example 3.3.
2. Elman, Karpenko & Merkurjev (2008), Proposition 22.9.
3. Harris (1995), Theorem 22.13.
4. Elman, Karpenko, & Merkurjev (2008), Proposition 7.28.
5. Harris (1995), Theorem 22.14.
6. Harris (1995), Lecture 22, p. 284.
7. Harris (1995), Lecture 22, p. 285.
8. Harris (1995), Exercise 22.6.
9. Harris (1995), Example 22.7.
10. Harris (1995), Theorem 22.14.
11. Fulton (1998), Example 19.1.11.
12. Elman, Karpenko & Merkurjev (2008), Proposition 68.1.
13. Elman, Karpenko, & Merkurjev (2008), Exercise 68.3.
14. Ottaviani (1988), section 1.
15. Mimura & Toda (1991), Theorem III.6.11.
16. Kapranov (1988), Theorem 4.10.
17. Swan (1985), Theorem 1.
References
• Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander (2008), Algebraic and geometric theory of quadratic forms, American Mathematical Society, ISBN 978-0-8218-4329-1, MR 2427530
• Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
• Harris, Joe (1995), Algebraic geometry: a first course, Springer-Verlag, ISBN 0-387-97716-3, MR 1416564
• Kapranov, Mikhail (1988), "On the derived categories of coherent sheaves on some homogeneous spaces", Inventiones Mathematicae, 92 (3): 479–508, Bibcode:1988InMat..92..479K, doi:10.1007/BF01393744, MR 0939472, S2CID 119584668
• Mimura, Mamoru; Toda, Hirosi (1992), Topology of Lie groups, American Mathematical Society, ISBN 978-0821813423, MR 1122592
• Ottaviani, Giorgio (1988), "Spinor bundles on quadrics", Transactions of the American Mathematical Society, 307: 301–316, doi:10.1090/S0002-9947-1988-0936818-5, MR 0936818
• Swan, Richard (1985), "K-theory of quadric hypersurfaces", Annals of Mathematics, 122 (1): 113–153, doi:10.2307/1971371, JSTOR 1971371, MR 0799254
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Wikipedia
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Quadric geometric algebra
Quadric geometric algebra (QGA) is a geometrical application of the ${\mathcal {G}}_{6,3}$ geometric algebra. This algebra is also known as the ${\mathcal {C}}\ell _{6,3}$ Clifford algebra. QGA is a super-algebra over ${\mathcal {G}}_{4,1}$ conformal geometric algebra (CGA) and ${\mathcal {G}}_{1,3}$ spacetime algebra (STA), which can each be defined within sub-algebras of QGA.
CGA provides representations of spherical entities (points, spheres, planes, and lines) and a complete set of operations (translation, rotation, dilation, and intersection) that apply to them. QGA extends CGA to also include representations of some non-spherical entities: principal axes-aligned quadric surfaces and many of their degenerate forms such as planes, lines, and points.
General quadric surfaces are characterized by the implicit polynomial equation of degree 2
$Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fzx+Gx+Hy+Iz+J=0.$
which can characterize quadric surfaces located at any center point and aligned along arbitrary axes. However, QGA includes vector entities that can represent only the principal axes-aligned quadric surfaces characterized by
$Ax^{2}+By^{2}+Cz^{2}+Gx+Hy+Iz+J=0.$
This is still a very significant advancement over CGA.
A possible performance issue with using QGA is the increased computation required to use a 9D vector space, as compared to the smaller 5D vector space of CGA. A 5D CGA subspace can be used when only CGA entities are involved in computations.
In general, the operation of rotation does not work correctly on non-spherical QGA quadric surface entities. Rotation also does not work correctly on the QGA point entities. Attempting to rotate a QGA quadric surface may result in a different type of quadric surface, or a quadric surface that is rotated and distorted in an unexpected way. Attempting to rotate a QGA point may produce a value that projects as the expected rotated vector, but the produced value is generally not a correct embedding of the rotated vector. The failure of QGA points to rotate correctly also leads to the inability to use outermorphisms to rotate dual Geometric Outer Product Null Space (GOPNS) entities. To rotate a QGA point, it must be projected to a vector or converted to a CGA point for rotation operations, then the rotated result can be re-embedded or converted back into a QGA point. A quadric surface rotated by an arbitrary angle cannot be represented by any known QGA entity. Representation of general quadric surfaces with useful operations will require an algebra (that appears to be unknown at this time) that extends QGA.
Although rotation is generally unavailable in QGA, the transposition operation is a special-case modification of rotation by $\pi /2$ that works correctly on all QGA GIPNS entities. Transpositions allow QGA GIPNS entities to be reflected in the six diagonal planes $y=\pm x$, $z=\pm x$, and $z=\pm y$.
Entities for all principal axes-aligned quadric surfaces can be defined in QGA. These include ellipsoids, cylinders, cones, paraboloids, and hyperboloids in all of their various forms.
A powerful feature of QGA is the ability to compute the intersections of axes-aligned quadric surfaces. With few exceptions, the outer product of QGA GIPNS surface entities represents their surfaces intersection(s). This method of computing intersections works the same as it does in CGA, where only spherical entities are available.
References
Julio Zamora-Esquivel. "G6,3 Geometric Algebra; Description and Implementation." Advances in Applied Clifford Algebras 24 (2014), 493-514. Springer Basel.
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Wikipedia
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Line complex
In algebraic geometry, a line complex is a 3-fold given by the intersection of the Grassmannian G(2, 4) (embedded in projective space P5 by Plücker coordinates) with a hypersurface. It is called a line complex because points of G(2, 4) correspond to lines in P3, so a line complex can be thought of as a 3-dimensional family of lines in P3. The linear line complex and quadric line complex are the cases when the hypersurface has degree 1 or 2; they are both rational varieties.
References
• Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523
• Jessop, C. M. (2001) [1903], A treatise on the line complex, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2913-4, MR 0247995
• Klein, Felix (1870), "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen, Springer Berlin / Heidelberg, 2 (2): 198–226, doi:10.1007/BF01444020, ISSN 0025-5831, S2CID 121706710
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Wikipedia
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Multinomial theorem
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.
Theorem
For any positive integer m and any non-negative integer n, the multinomial formula describes how a sum with m terms expands when raised to an arbitrary power n:
$(x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n;\ k_{1},k_{2},\cdots ,k_{m}\geq 0}{n \choose k_{1},k_{2},\ldots ,k_{m}}\prod _{t=1}^{m}x_{t}^{k_{t}}\,,$
where
${n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}$
is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices k1 through km such that the sum of all ki is n. That is, for each term in the expansion, the exponents of the xi must add up to n. Also, as with the binomial theorem, quantities of the form x0 that appear are taken to equal 1 (even when x equals zero).
In the case m = 2, this statement reduces to that of the binomial theorem.
Example
The third power of the trinomial a + b + c is given by
$(a+b+c)^{3}=a^{3}+b^{3}+c^{3}+3a^{2}b+3a^{2}c+3b^{2}a+3b^{2}c+3c^{2}a+3c^{2}b+6abc.$
This can be computed by hand using the distributive property of multiplication over addition, but it can also be done (perhaps more easily) with the multinomial theorem. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example:
$a^{2}b^{0}c^{1}$ has the coefficient ${3 \choose 2,0,1}={\frac {3!}{2!\cdot 0!\cdot 1!}}={\frac {6}{2\cdot 1\cdot 1}}=3.$
$a^{1}b^{1}c^{1}$ has the coefficient ${3 \choose 1,1,1}={\frac {3!}{1!\cdot 1!\cdot 1!}}={\frac {6}{1\cdot 1\cdot 1}}=6.$
Alternate expression
The statement of the theorem can be written concisely using multiindices:
$(x_{1}+\cdots +x_{m})^{n}=\sum _{|\alpha |=n}{n \choose \alpha }x^{\alpha }$
where
$\alpha =(\alpha _{1},\alpha _{2},\dots ,\alpha _{m})$
and
$x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{m}^{\alpha _{m}}$
Proof
This proof of the multinomial theorem uses the binomial theorem and induction on m.
First, for m = 1, both sides equal x1n since there is only one term k1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m. Then
${\begin{aligned}&(x_{1}+x_{2}+\cdots +x_{m}+x_{m+1})^{n}=(x_{1}+x_{2}+\cdots +(x_{m}+x_{m+1}))^{n}\\[6pt]={}&\sum _{k_{1}+k_{2}+\cdots +k_{m-1}+K=n}{n \choose k_{1},k_{2},\ldots ,k_{m-1},K}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}(x_{m}+x_{m+1})^{K}\end{aligned}}$
by the induction hypothesis. Applying the binomial theorem to the last factor,
$=\sum _{k_{1}+k_{2}+\cdots +k_{m-1}+K=n}{n \choose k_{1},k_{2},\ldots ,k_{m-1},K}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}\sum _{k_{m}+k_{m+1}=K}{K \choose k_{m},k_{m+1}}x_{m}^{k_{m}}x_{m+1}^{k_{m+1}}$
$=\sum _{k_{1}+k_{2}+\cdots +k_{m-1}+k_{m}+k_{m+1}=n}{n \choose k_{1},k_{2},\ldots ,k_{m-1},k_{m},k_{m+1}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m-1}^{k_{m-1}}x_{m}^{k_{m}}x_{m+1}^{k_{m+1}}$
which completes the induction. The last step follows because
${n \choose k_{1},k_{2},\ldots ,k_{m-1},K}{K \choose k_{m},k_{m+1}}={n \choose k_{1},k_{2},\ldots ,k_{m-1},k_{m},k_{m+1}},$
as can easily be seen by writing the three coefficients using factorials as follows:
${\frac {n!}{k_{1}!k_{2}!\cdots k_{m-1}!K!}}{\frac {K!}{k_{m}!k_{m+1}!}}={\frac {n!}{k_{1}!k_{2}!\cdots k_{m+1}!}}.$
Multinomial coefficients
The numbers
${n \choose k_{1},k_{2},\ldots ,k_{m}}$
appearing in the theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials:
${n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}={k_{1} \choose k_{1}}{k_{1}+k_{2} \choose k_{2}}\cdots {k_{1}+k_{2}+\cdots +k_{m} \choose k_{m}}$
Sum of all multinomial coefficients
The substitution of xi = 1 for all i into the multinomial theorem
$\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{n \choose k_{1},k_{2},\ldots ,k_{m}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}}=(x_{1}+x_{2}+\cdots +x_{m})^{n}$
gives immediately that
$\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{n \choose k_{1},k_{2},\ldots ,k_{m}}=m^{n}.$
Number of multinomial coefficients
The number of terms in a multinomial sum, #n,m, is equal to the number of monomials of degree n on the variables x1, …, xm:
$\#_{n,m}={n+m-1 \choose m-1}.$
The count can be performed easily using the method of stars and bars.
Valuation of multinomial coefficients
The largest power of a prime p that divides a multinomial coefficient may be computed using a generalization of Kummer's theorem.
Interpretations
Ways to put objects into bins
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k1 objects in the first bin, k2 objects in the second bin, and so on.[1]
Number of ways to select according to a distribution
In statistical mechanics and combinatorics, if one has a number distribution of labels, then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution {ni} on a set of N total items, ni represents the number of items to be given the label i. (In statistical mechanics i is the label of the energy state.)
The number of arrangements is found by
• Choosing n1 of the total N to be labeled 1. This can be done ${\tbinom {N}{n_{1}}}$ ways.
• From the remaining N − n1 items choose n2 to label 2. This can be done ${\tbinom {N-n_{1}}{n_{2}}}$ ways.
• From the remaining N − n1 − n2 items choose n3 to label 3. Again, this can be done ${\tbinom {N-n_{1}-n_{2}}{n_{3}}}$ ways.
Multiplying the number of choices at each step results in:
${N \choose n_{1}}{N-n_{1} \choose n_{2}}{N-n_{1}-n_{2} \choose n_{3}}\cdots ={\frac {N!}{(N-n_{1})!n_{1}!}}\cdot {\frac {(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}}\cdot {\frac {(N-n_{1}-n_{2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}}\cdots .$
Cancellation results in the formula given above.
Number of unique permutations of words
The multinomial coefficient
${\binom {n}{k_{1},\ldots ,k_{m}}}$
is also the number of distinct ways to permute a multiset of n elements, where ki is the multiplicity of each of the ith element. For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps, is
${11 \choose 1,4,4,2}={\frac {11!}{1!\,4!\,4!\,2!}}=34650.$
Generalized Pascal's triangle
One can use the multinomial theorem to generalize Pascal's triangle or Pascal's pyramid to Pascal's simplex. This provides a quick way to generate a lookup table for multinomial coefficients.
See also
• Multinomial distribution
• Stars and bars (combinatorics)
References
1. National Institute of Standards and Technology (May 11, 2010). "NIST Digital Library of Mathematical Functions". Section 26.4. Retrieved August 30, 2010.
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Wikipedia
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Rectified 9-cubes
In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube.
9-orthoplex
Rectified 9-orthoplex
Birectified 9-orthoplex
Trirectified 9-orthoplex
Quadrirectified 9-cube
Trirectified 9-cube
Birectified 9-cube
Rectified 9-cube
9-cube
Orthogonal projections in BC9 Coxeter plane
There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-cube are located in the square face centers of the 9-cube. Vertices of the trirectified 9-orthoplex are located in the cube cell centers of the 9-cube. Vertices of the quadrirectified 9-cube are located in the tesseract centers of the 9-cube.
These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.
Rectified 9-cube
Alternate names
• Rectified enneract (Acronym ren) (Jonathan Bowers)[1]
Images
orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
— — —
[8] [6] [4]
Birectified 9-cube
Alternate names
• Birectified enneract (Acronym barn) (Jonathan Bowers)[2]
Images
orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
— — —
[8] [6] [4]
Trirectified 9-cube
Alternate names
• Trirectified enneract (Acronym tarn) (Jonathan Bowers)[3]
Images
orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
— — —
[8] [6] [4]
Quadrirectified 9-cube
Alternate names
• Quadrirectified enneract (Acronym nav) (Jonathan Bowers)[4]
Images
orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
— — —
[8] [6] [4]
Notes
1. Klitzing (o3o3o3o3o3o3o3x4o - ren)
2. Klitzing (o3o3o3o3o3o3x3o4o - barn)
3. Klitzing (o3o3o3o3o3x3o3o4o - tarn)
4. Klitzing (o3o3o3o3x3o3o3o4o - nav)
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "9D uniform polytopes (polyyotta)". x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne
External links
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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Wikipedia
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Quadrisecant
In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curve can have with a line, and for such curves the quadrisecants form a discrete set of lines. Quadrisecants have been studied for curves of several types:
• Knots and links in knot theory, when nontrivial, always have quadrisecants, and the existence and number of quadrisecants has been studied in connection with knot invariants including the minimum total curvature and the ropelength of a knot.
• The number of quadrisecants of a non-singular algebraic curve in complex projective space can be computed by a formula derived by Arthur Cayley.
• Quadrisecants of arrangements of skew lines touch subsets of four lines from the arrangement. They are associated with ruled surfaces and the Schläfli double six configuration.
Definition and motivation
A quadrisecant is a line that intersects a curve, surface, or other set in four distinct points. It is analogous to a secant line, a line that intersects a curve or surface in two points; and a trisecant, a line that intersects a curve or surface in three points.[2]
Compared to secants and trisecants, quadrisecants are especially relevant for space curves, because they have the largest possible number of intersection points of a line with a generic curve. In the plane, a generic curve can be crossed arbitrarily many times by a line; for instance, small generic perturbations of the sine curve are crossed infinitely often by the horizontal axis. In contrast, if an arbitrary space curve is perturbed by a small distance to make it generic, there will be no lines through five or more points of the perturbed curve. Nevertheless, any quadrisecants of the original space curve will remain present nearby in its perturbation.[3]
One explanation for this phenomenon is visual: looking at a space curve from far away, the space of such points of view can be described as a two-dimensional sphere, one point corresponding to each direction. Pairs of strands of the curve may appear to cross from all of these points of view, or from a two-dimensional subset of them. Three strands will form a triple crossing when the point of view lies on a trisecant, and four strands will form a quadruple crossing from a point of view on a quadrisecant. Each constraint that the crossing of a pair of strands lies on another strand reduces the number of degrees of freedom by one (for a generic curve), so the points of view on trisecants form a one-dimensional (continuously infinite) subset of the sphere, while the points of view on quadrisecants form a zero-dimensional (discrete) subset. C. T. C. Wall writes that the fact that generic space curves are crossed at most four times by lines is "one of the simplest theorems of the kind", a model case for analogous theorems on higher-dimensional transversals.[3]
Additionally, for generic space curves, the quadrisecants form a discrete set of lines that in contrast to the trisecants which, when they occur, form continuous families of lines.[4] Depending on the properties of the curve, it may have no quadrisecants, finitely many, or infinitely many. These considerations make it of interest to determine conditions for the existence of quadrisecants, or to find bounds on their number in various special cases, such as knotted curves,[5][6] algebraic curves,[7] or arrangements of lines.[8]
For special classes of curves
Knots and links
In three-dimensional Euclidean space, every nontrivial tame knot or link has a quadrisecant. Originally established in the case of knotted polygons and smooth knots by Erika Pannwitz,[5] this result was extended to knots in suitably general position and links with nonzero linking number,[6] and later to all nontrivial tame knots and links.[9]
Pannwitz proved more strongly that, for a locally flat disk having the knot as its boundary, the number of singularities of the disk can be used to construct a lower bound on the number of distinct quadrisecants. The existence of at least one quadrisecant follows from the fact that any such disk must have at least one singularity.[5][10] Morton & Mond (1982) conjectured that the number of distinct quadrisecants of a given knot is always at least $n(n-1)/2$, where $n$ is the crossing number of the knot.[6][10] Counterexamples to this conjecture have since been discovered.[10]
Two-component links have quadrisecants in which the points on the quadrisecant appear in alternating order between the two components,[6] and nontrivial knots have quadrisecants in which the four points, ordered cyclically as $abcd$ on the knot, appear in order $acbd$ along the quadrisecant.[11] The existence of these alternating quadrisecants can be used to derive the Fáry–Milnor theorem, a lower bound on the total curvature of a nontrivial knot.[11] Quadrisecants have also been used to find lower bounds on the ropelength of knots.[12]
G. T. Jin and H. S. Kim conjectured that, when a knotted curve $K$ has finitely many quadrisecants, $K$ can be approximated with an equivalent polygonal knot with its vertices at the points where the quadrisecants intersect $K$, in the same order as they appear on $K$. However, their conjecture is false: in fact, for every knot type, there is a realization for which this construction leads to a self-intersecting polygon, and another realization where this construction produces a knot of a different type.[13]
Unsolved problem in mathematics:
Does every wild knot have infinitely many quadrisecants?
(more unsolved problems in mathematics)
It has been conjectured that every wild knot has an infinite number of quadrisecants.[9]
Algebraic curves
Arthur Cayley derived a formula for the number of quadrisecants of an algebraic curve in three-dimensional complex projective space, as a function of its degree and genus.[7] For a curve of degree $d$ and genus $g$, the number of quadrisecants is[14]
${\frac {(d-2)(d-3)^{2}(d-4)}{12}}-{\frac {g(d^{2}-7d+13-g)}{2}}.$
This formula assumes that the given curve is non-singular; adjustments may be necessary if it has singular points.[15][16]
Skew lines
In three-dimensional Euclidean space, every set of four skew lines in general position has either two quadrisecants (also in this context called transversals) or none. Any three of the four lines determine a hyperboloid, a doubly ruled surface in which one of the two sets of ruled lines contains the three given lines, and the other ruling consists of trisecants to the given lines. If the fourth of the given lines pierces this surface, it has two points of intersection, because the hyperboloid is defined by a quadratic equation. The two trisecants of the ruled surface, through these two points, form two quadrisecants of the given four lines. On the other hand, if the fourth line is disjoint from the hyperboloid, then there are no quadrisecants.[17] In spaces with complex number coordinates rather than real coordinates, four skew lines always have exactly two quadrisecants.[8]
The quadrisecants of sets of lines play an important role in the construction of the Schläfli double six, a configuration of twelve lines intersecting each other in 30 crossings. If five lines $a_{i}$ (for $i=1,2,3,4,5$) are given in three-dimensional space, such that all five are intersected by a common line $b_{6}$ but are otherwise in general position, then each of the five quadruples of the lines $a_{i}$ has a second quadrisecant $b_{i}$, and the five lines $b_{i}$ formed in this way are all intersected by a common line $a_{6}$. These twelve lines and the 30 intersection points $a_{i}b_{j}$ form the double six.[18][19]
An arrangement of $n$ complex lines with a given number of pairwise intersections and otherwise skew may be interpreted as an algebraic curve with degree $n$ and with genus determined from its number of intersections, and Cayley's aforementioned formula used to count its quadrisecants. The same result as this formula can also be obtained by classifying the quadruples of lines by their intersections, counting the number of quadrisecants for each type of quadruple, and summing over all quadruples of lines in the given set.[8]
References
1. Jin, Gyo Taek (December 2017), "Polygonal approximation of unknots by quadrisecants", in Reiter, Philipp; Blatt, Simon; Schikorra, Armin (eds.), New Directions in Geometric and Applied Knot Theory, De Gruyter Open, pp. 159–175, doi:10.1515/9783110571493-008
2. Eisenbud, David; Harris, Joe (2016), 3264 and All That: A second course in algebraic geometry, Cambridge, UK: Cambridge University Press, p. 377, doi:10.1017/CBO9781139062046, ISBN 978-1-107-60272-4, MR 3617981
3. Wall, C. T. C. (1977), "Geometric properties of generic differentiable manifolds", in Palis, Jacob; do Carmo, Manfredo (eds.), Geometry and Topology: Proceedings of the Latin American School of Mathematics (ELAM III) held at the Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, July 1976, Lecture Notes in Mathematics, vol. 597, pp. 707–774, doi:10.1007/BFb0085382, MR 0494233
4. Denne, Elizabeth (2018), "Quadrisecants and essential secants of knots", in Blatt, Simon; Reiter, Philipp; Schikorra, Armin (eds.), New directions in geometric and applied knot theory, Partial Differential Equations and Measure Theory, De Gruyter, Berlin, pp. 138–158, doi:10.1515/9783110571493-006, MR 3915943, S2CID 128222971
5. Pannwitz, Erika (1933), "Eine elementargeometrische Eigenschaft von Verschlingungen und Knoten", Mathematische Annalen, 108 (1): 629–672, doi:10.1007/BF01452857, S2CID 123026724
6. Morton, Hugh R.; Mond, David M. Q. (1982), "Closed curves with no quadrisecants", Topology, 21 (3): 235–243, doi:10.1016/0040-9383(82)90007-6, MR 0649756
7. Cayley, Arthur (1863), Philosophical Transactions of the Royal Society of London, vol. 153, The Royal Society, pp. 453–483, JSTOR 108806
8. Wong, B. C. (1934), "Enumerative properties of $r$-space curves", Bulletin of the American Mathematical Society, 40 (4): 291–296, doi:10.1090/S0002-9904-1934-05854-3, MR 1562839
9. Kuperberg, Greg (1994), "Quadrisecants of knots and links", Journal of Knot Theory and Its Ramifications, 3: 41–50, arXiv:math/9712205, doi:10.1142/S021821659400006X, MR 1265452, S2CID 6103528
10. Jin, Gyo Taek (2005), "Quadrisecants of knots with small crossing number", Physical and numerical models in knot theory (PDF), Ser. Knots Everything, vol. 36, Singapore: World Scientific Publishing, pp. 507–523, doi:10.1142/9789812703460_0025, MR 2197955
11. Denne, Elizabeth Jane (2004), Alternating quadrisecants of knots, Ph.D. thesis, University of Illinois at Urbana-Champaign, arXiv:math/0510561, Bibcode:2005math.....10561D
12. Denne, Elizabeth; Diao, Yuanan; Sullivan, John M. (2006), "Quadrisecants give new lower bounds for the ropelength of a knot", Geometry & Topology, 10: 1–26, arXiv:math/0408026, doi:10.2140/gt.2006.10.1, MR 2207788, S2CID 5770206
13. Bai, Sheng; Wang, Chao; Wang, Jiajun (2018), "Counterexamples to the quadrisecant approximation conjecture", Journal of Knot Theory and Its Ramifications, 27 (2), 1850022, arXiv:1605.00538, doi:10.1142/S0218216518500220, MR 3770471, S2CID 119601013
14. Griffiths, Phillip; Harris, Joseph (2011), Principles of Algebraic Geometry, Wiley Classics Library, vol. 52, John Wiley & Sons, p. 296, ISBN 9781118030776
15. Welchman, W. G. (April 1932), "Note on the trisecants and quadrisecants of a space curve", Mathematical Proceedings of the Cambridge Philosophical Society, 28 (2): 206–208, doi:10.1017/s0305004100010872, S2CID 120725025
16. Maxwell, Edwin A. (July 1935), "Note on the formula for the number of quadrisecants of a curve in space of three dimensions", Mathematical Proceedings of the Cambridge Philosophical Society, 31 (3): 324–326, doi:10.1017/s0305004100013086, S2CID 122279811
17. Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, p. 164, ISBN 978-0-8284-1087-8
18. Schläfli, Ludwig (1858), Cayley, Arthur (ed.), "An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface", Quarterly Journal of Pure and Applied Mathematics, 2: 55–65, 110–120
19. Coxeter, H. S. M. (2006), "An absolute property of four mutually tangent circles", Non-Euclidean geometries, Math. Appl. (N. Y.), vol. 581, New York: Springer, pp. 109–114, doi:10.1007/0-387-29555-0_5, MR 2191243; Coxeter repeats Schläfli's construction, and provides several references to simplified proofs of its correctness
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Wikipedia
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Qualitative economics
Qualitative economics is the representation and analysis of information about the direction of change (+, -, or 0) in some economic variable(s) as related to change of some other economic variable(s). For the non-zero case, what makes the change qualitative is that its direction but not its magnitude is specified.[1]
Typical exercises of qualitative economics include comparative-static changes studied in microeconomics or macroeconomics and comparative equilibrium-growth states in a macroeconomic growth model. A simple example illustrating qualitative change is from macroeconomics. Let:
GDP = nominal gross domestic product, a measure of national income
M = money supply
T = total taxes.
Monetary theory hypothesizes a positive relationship between GDP the dependent variable and M the independent variable. Equivalent ways to represent such a qualitative relationship between them are as a signed functional relationship and as a signed derivative:
$GDP=f({\overset {+}{M}})\quad \!$ or $\quad {\frac {df(M)}{dM}}>0.$
where the '+' indexes a positive relationship of GDP to M, that is, as M increases, GDP increases as a result.
Another model of GDP hypothesizes that GDP has a negative relationship to T. This can be represented similarly to the above, with a theoretically appropriate sign change as indicated:
$GDP=f({\overset {-}{T}})\quad \!$ or $\quad {\frac {df(T)}{dT}}<0.$
That is, as T increases, GDP decreases as a result. A combined model uses both M and T as independent variables. The hypothesized relationships can be equivalently represented as signed functional relationships and signed partial derivatives (suitable for more than one independent variable):
$GDP=f({\overset {+}{M}},{\overset {-}{T}})\,\!\quad $ or $\quad {\frac {\partial f(M,T)}{\partial M}}>0,\quad $ ${\frac {\partial f(M,T)}{\partial T}}<0.$
Qualitative hypotheses occur in earliest history of formal economics but only as to formal economic models from the late 1930s with Hicks's model of general equilibrium in a competitive economy.[2] A classic exposition of qualitative economics is Samuelson, 1947.[3] There Samuelson identifies qualitative restrictions and the hypotheses of maximization and stability of equilibrium as the three fundamental sources of meaningful theorems — hypotheses about empirical data that could conceivably be refuted by empirical data.[1]
Notes
1. James Quirk, 1987. "qualitative economics," The New Palgrave: A Dictionary of Economics, v. 4, p. 1.
2. J. R. Hicks, 1939. Value and Capital. Oxford.
3. Paul A. Samuelson, 1947. Foundations of Economic Analysis, pp. 5, 21-29.
References
• J. R. Hicks, 1939. Value and Capital. Oxford.
• Kelvin Lancaster, 1962. "The Scope of Qualitative Economics," Review of Economic Studies, 29(2), pp. 99-123.
• W.M. Gorman, 1964. "More Scope for Qualitative Economics," Review of Economic Studies, 31(1) pp. 65-68.
• James Quirk, 1987. "qualitative economics," The New Palgrave: A Dictionary of Economics, v. 4, pp. 1-3.
• _____ and Richard Ruppert, 1965. "Qualitative Economics and the Stability of Equilibrium," Review of Economic Studies, 32(4), pp. 311-326.
• Paul A. Samuelson, 1947. Foundations of Economic Analysis, Harvard University Press. ISBN 0-674-31301-1
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Wikipedia
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Entitative graph
An entitative graph is an element of the diagrammatic syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic beginning in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. See 3.468, 4.434, and 4.564 in Peirce's Collected Papers. Peirce wrote of this system in an 1897 Monist article titled "The Logic of Relatives", although he had mentioned logical graphs in an 1882 letter to O. H. Mitchell.[1]
The syntax is:
• The blank page;
• Single letters, phrases;
• Dashes;
• Objects (subgraphs) enclosed by a simple closed curve called a cut. A cut can be empty.
The semantics are:
• The blank page denotes False;
• Letters, phrases, subgraphs, and entire graphs can be True or False;
• To surround objects with a cut is equivalent to Boolean complementation.[1] Hence an empty cut denotes Truth;
• All objects within a given cut are tacitly joined by disjunction.[1]
• A dash is read "everything" if it is encircled an even number of times, and read "something" if it is encircled an odd number of times.[1]
Entitative graphs are read from outside to inside.[1]
A "proof" manipulates a graph, using a short list of rules, until the graph is reduced to an empty cut or the blank page. A graph that can be so reduced is what is now called a tautology (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of the satisfiable formulas of first-order logic.
Peirce soon abandoned the entitative graphs for the existential graphs, whose sentential (alpha) part is dual to the entitative graphs. He developed the existential graphs until they became another formalism for what are now termed first-order logic and normal modal logic.
The primary algebra of G. Spencer-Brown's Laws of Form is isomorphic to the entitative graphs.
See also
• Charles Sanders Peirce bibliography
References
1. Hawkins 1975, p. 129.
Bibliography
• Hawkins, Benjamin S. (1975). "The Existential Graphs of Charles S. Peirce". Transactions of the Charles S.Peirce Society. 11 (2): 128–139. JSTOR 40319733. ProQuest 1303699159 – via ProQuest.
• Peirce, C. S., Collected Papers of Charles Sanders Peirce, Vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), Vols. 7–8, Arthur W. Burks, ed., Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as CP volume.paragraph.
• Peirce, C. S., "Qualitative Logic", MS 736 (c. 1886), pp. 101–115 in The New Elements of Mathematics by Charles S. Peirce, Volume 4, Mathematical Philosophy, Carolyn Eisele (ed.), Mouton, The Hague, 1976.
• Peirce, C. S., "Qualitative Logic", MS 582 (1886), pp. 323–371 in Writings of Charles S. Peirce: A Chronological Edition, Volume 5, 1884–1886, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
• Peirce, C. S., "The Logic of Relatives: Qualitative and Quantitative", MS 584 (1886), pp. 372–378 in Writings of Charles S. Peirce: A Chronological Edition, Volume 5, 1884–1886, Peirce Edition Project (eds.), Indiana University Press, Bloomington, IN, 1993.
• Shin, Sun-Joo (2002), The Iconic Logic of Peirce's Graphs, MIT Press, Cambridge, MA.
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Wikipedia
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Discrete choice
In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case, calculus methods (e.g. first-order conditions) can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis. On the other hand, discrete choice analysis examines situations in which the potential outcomes are discrete, such that the optimum is not characterized by standard first-order conditions. Thus, instead of examining "how much" as in problems with continuous choice variables, discrete choice analysis examines "which one". However, discrete choice analysis can also be used to examine the chosen quantity when only a few distinct quantities must be chosen from, such as the number of vehicles a household chooses to own [1] and the number of minutes of telecommunications service a customer decides to purchase.[2] Techniques such as logistic regression and probit regression can be used for empirical analysis of discrete choice.
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Discrete choice models theoretically or empirically model choices made by people among a finite set of alternatives. The models have been used to examine, e.g., the choice of which car to buy,[1][3] where to go to college,[4] which mode of transport (car, bus, rail) to take to work[5] among numerous other applications. Discrete choice models are also used to examine choices by organizations, such as firms or government agencies. In the discussion below, the decision-making unit is assumed to be a person, though the concepts are applicable more generally. Daniel McFadden won the Nobel prize in 2000 for his pioneering work in developing the theoretical basis for discrete choice.
Discrete choice models statistically relate the choice made by each person to the attributes of the person and the attributes of the alternatives available to the person. For example, the choice of which car a person buys is statistically related to the person's income and age as well as to price, fuel efficiency, size, and other attributes of each available car. The models estimate the probability that a person chooses a particular alternative. The models are often used to forecast how people's choices will change under changes in demographics and/or attributes of the alternatives.
Discrete choice models specify the probability that an individual chooses an option among a set of alternatives. The probabilistic description of discrete choice behavior is used not to reflect individual behavior that is viewed as intrinsically probabilistic. Rather, it is the lack of information that leads us to describe choice in a probabilistic fashion. In practice, we cannot know all factors affecting individual choice decisions as their determinants are partially observed or imperfectly measured. Therefore, discrete choice models rely on stochastic assumptions and specifications to account for unobserved factors related to a) choice alternatives, b) taste variation over people (interpersonal heterogeneity) and over time (intra-individual choice dynamics), and c) heterogeneous choice sets. The different formulations have been summarized and classified into groups of models.[6]
Applications
• Marketing researchers use discrete choice models to study consumer demand and to predict competitive business responses, enabling choice modelers to solve a range of business problems, such as pricing, product development, and demand estimation problems. In market research, this is commonly called conjoint analysis.[1]
• Transportation planners use discrete choice models to predict demand for planned transportation systems, such as which route a driver will take and whether someone will take rapid transit systems.[5][7] The first applications of discrete choice models were in transportation planning, and much of the most advanced research in discrete choice models is conducted by transportation researchers.
• Energy forecasters and policymakers use discrete choice models for households' and firms' choice of heating system, appliance efficiency levels, and fuel efficiency level of vehicles.[8][9]
• Environmental studies utilize discrete choice models to examine the recreators' choice of, e.g., fishing or skiing site and to infer the value of amenities, such as campgrounds, fish stock, and warming huts, and to estimate the value of water quality improvements.[10]
• Labor economists use discrete choice models to examine participation in the work force, occupation choice, and choice of college and training programs.[4]
• Ecological studies employ discrete choice models to investigate parameters that drive habitat selection in animals.[11]
Common features of discrete choice models
Discrete choice models take many forms, including: Binary Logit, Binary Probit, Multinomial Logit, Conditional Logit, Multinomial Probit, Nested Logit, Generalized Extreme Value Models, Mixed Logit, and Exploded Logit. All of these models have the features described below in common.
Choice set
The choice set is the set of alternatives that are available to the person. For a discrete choice model, the choice set must meet three requirements:
1. The set of alternatives must be collectively exhaustive, meaning that the set includes all possible alternatives. This requirement implies that the person necessarily does choose an alternative from the set.
2. The alternatives must be mutually exclusive, meaning that choosing one alternative means not choosing any other alternatives. This requirement implies that the person chooses only one alternative from the set.
3. The set must contain a finite number of alternatives. This third requirement distinguishes discrete choice analysis from forms of regression analysis in which the dependent variable can (theoretically) take an infinite number of values.
As an example, the choice set for a person deciding which mode of transport to take to work includes driving alone, carpooling, taking bus, etc. The choice set is complicated by the fact that a person can use multiple modes for a given trip, such as driving a car to a train station and then taking train to work. In this case, the choice set can include each possible combination of modes. Alternatively, the choice can be defined as the choice of "primary" mode, with the set consisting of car, bus, rail, and other (e.g. walking, bicycles, etc.). Note that the alternative "other" is included in order to make the choice set exhaustive.
Different people may have different choice sets, depending on their circumstances. For instance, the Scion automobile was not sold in Canada as of 2009, so new car buyers in Canada faced different choice sets from those of American consumers. Such considerations are taken into account in the formulation of discrete choice models.
Defining choice probabilities
A discrete choice model specifies the probability that a person chooses a particular alternative, with the probability expressed as a function of observed variables that relate to the alternatives and the person. In its general form, the probability that person n chooses alternative i is expressed as:
$P_{ni}\equiv \Pr({\text{Person }}n{\text{ chooses alternative }}i)=G(x_{ni},\;x_{nj,j\neq i},\;s_{n},\;\beta ),$
where
$x_{ni}$ is a vector of attributes of alternative i faced by person n,
$x_{nj,j\neq i}$ is a vector of attributes of the other alternatives (other than i) faced by person n,
$s_{n}$ is a vector of characteristics of person n, and
$\beta $ is a set of parameters giving the effects of variables on probabilities, which are estimated statistically.
In the mode of transport example above, the attributes of modes (xni), such as travel time and cost, and the characteristics of consumer (sn), such as annual income, age, and gender, can be used to calculate choice probabilities. The attributes of the alternatives can differ over people; e.g., cost and time for travel to work by car, bus, and rail are different for each person depending on the location of home and work of that person.
Properties:
• Pni is between 0 and 1
• $\forall n:\;\sum _{j=1}^{J}P_{nj}=1,$ where J is the total number of alternatives.
• (Expected fraction of people choosing i ) $={1 \over N}{\sum _{n=1}^{N}P_{ni}},$ where N is the number of people making the choice.
Different models (i.e., models using a different function G) have different properties. Prominent models are introduced below.
Consumer utility
Discrete choice models can be derived from utility theory. This derivation is useful for three reasons:
1. It gives a precise meaning to the probabilities Pni
2. It motivates and distinguishes alternative model specifications, e.g., the choice of a functional form for G.
3. It provides the theoretical basis for calculation of changes in consumer surplus (compensating variation) from changes in the attributes of the alternatives.
Uni is the utility (or net benefit or well-being) that person n obtains from choosing alternative i. The behavior of the person is utility-maximizing: person n chooses the alternative that provides the highest utility. The choice of the person is designated by dummy variables, yni, for each alternative:
$y_{ni}={\begin{cases}1&U_{ni}>U_{nj}\quad \forall j\neq i\\0&{\text{otherwise}}\end{cases}}$
Consider now the researcher who is examining the choice. The person's choice depends on many factors, some of which the researcher observes and some of which the researcher does not. The utility that the person obtains from choosing an alternative is decomposed into a part that depends on variables that the researcher observes and a part that depends on variables that the researcher does not observe. In a linear form, this decomposition is expressed as
$U_{ni}=\beta z_{ni}+\varepsilon _{ni}$
where
• $z_{ni}$ is a vector of observed variables relating to alternative i for person n that depends on attributes of the alternative, xni, interacted perhaps with attributes of the person, sn, such that it can be expressed as $z_{ni}=z(x_{ni},s_{n})$ for some numerical function z,
• $\beta $ is a corresponding vector of coefficients of the observed variables, and
• $\varepsilon _{ni}$ captures the impact of all unobserved factors that affect the person's choice.
The choice probability is then
${\begin{aligned}P_{ni}&=\Pr(y_{ni}=1)\\&=\Pr \left(\bigcap _{j\neq i}U_{ni}>U_{nj},\right)\\&=\Pr \left(\bigcap _{j\neq i}\beta z_{ni}+\varepsilon _{ni}>\beta z_{nj}+\varepsilon _{nj},\right)\\&=\Pr \left(\bigcap _{j\neq i}\varepsilon _{nj}-\varepsilon _{ni}<\beta z_{ni}-\beta z_{nj},\right)\end{aligned}}$
Given β, the choice probability is the probability that the random terms, εnj − εni (which are random from the researcher's perspective, since the researcher does not observe them) are below the respective quantities $\forall j\neq i:\beta z_{ni}-\beta z_{nj}.$ Different choice models (i.e. different specifications of G) arise from different distributions of εni for all i and different treatments of β.
Only differences matter
The probability that a person chooses a particular alternative is determined by comparing the utility of choosing that alternative to the utility of choosing other alternatives:
$P_{ni}=\Pr(y_{ni}=1)=\Pr \left(\bigcap _{j\neq i}U_{ni}>U_{nj}\right)=\Pr \left(\bigcap _{j\neq i}U_{ni}-U_{nj}>0\right)$
As the last term indicates, the choice probability depends only on the difference in utilities between alternatives, not on the absolute level of utilities. Equivalently, adding a constant to the utilities of all the alternatives does not change the choice probabilities.
Scale must be normalized
Since utility has no units, it is necessary to normalize the scale of utilities. The scale of utility is often defined by the variance of the error term in discrete choice models. This variance may differ depending on the characteristics of the dataset, such as when or where the data are collected. Normalization of the variance therefore affects the interpretation of parameters estimated across diverse datasets.
Prominent types of discrete choice models
Discrete choice models can first be classified according to the number of available alternatives.
* Binomial choice models (dichotomous): 2 available alternatives
* Multinomial choice models (polytomous): 3 or more available alternatives
Multinomial choice models can further be classified according to the model specification:
* Models, such as standard logit, that assume no correlation in unobserved factors over alternatives
* Models that allow correlation in unobserved factors among alternatives
In addition, specific forms of the models are available for examining rankings of alternatives (i.e., first choice, second choice, third choice, etc.) and for ratings data.
Details for each model are provided in the following sections.
Binary choice
Further information: binary regression
A. Logit with attributes of the person but no attributes of the alternatives
Further information: Logistic regression
Un is the utility (or net benefit) that person n obtains from taking an action (as opposed to not taking the action). The utility the person obtains from taking the action depends on the characteristics of the person, some of which are observed by the researcher and some are not. The person takes the action, yn = 1, if Un > 0. The unobserved term, εn, is assumed to have a logistic distribution. The specification is written succinctly as:
${\begin{cases}U_{n}=\beta s_{n}+\varepsilon _{n}\\y_{n}={\begin{cases}1&U_{n}>0\\0&U_{n}\leqslant 0\end{cases}}\\\varepsilon \sim {\text{Logistic}}\end{cases}}\quad \Rightarrow \quad P_{n1}={\frac {1}{1+\exp(-\beta s_{n})}}$
B. Probit with attributes of the person but no attributes of the alternatives
The description of the model is the same as model A, except the unobserved terms are distributed standard normal instead of logistic.
${\begin{cases}U_{n}=\beta s_{n}+\varepsilon _{n}\\y_{n}={\begin{cases}1&U_{n}>0\\0&U_{n}\leqslant 0\end{cases}}\\\varepsilon \sim {\text{Standard normal}}\end{cases}}\quad \Rightarrow \quad P_{n1}=\Phi (\beta s_{n}),$
where $\Phi $ is cumulative distribution function of standard normal.
C. Logit with variables that vary over alternatives
Uni is the utility person n obtains from choosing alternative i. The utility of each alternative depends on the attributes of the alternatives interacted perhaps with the attributes of the person. The unobserved terms are assumed to have an extreme value distribution.[nb 1]
${\begin{cases}U_{n1}=\beta z_{n1}+\varepsilon _{n1}\\U_{n2}=\beta z_{n2}+\varepsilon _{n2}\\\varepsilon _{n1},\varepsilon _{n2}\sim {\text{iid extreme value}}\end{cases}}\quad \Rightarrow \quad P_{n1}={\frac {\exp(\beta z_{n1})}{\exp(\beta z_{n1})+\exp(\beta z_{n2})}}$
We can relate this specification to model A above, which is also binary logit. In particular, Pn1 can also be expressed as
$P_{n1}={\frac {1}{1+\exp(-\beta (z_{n1}-z_{n2}))}}$
Note that if two error terms are iid extreme value,[nb 1] their difference is distributed logistic, which is the basis for the equivalence of the two specifications.
D. Probit with variables that vary over alternatives
The description of the model is the same as model C, except the difference of the two unobserved terms are distributed standard normal instead of logistic.
Then the probability of taking the action is
$P_{n1}=\Phi (\beta (z_{n1}-z_{n2})),$
where Φ is the cumulative distribution function of standard normal.
E. Logit with attributes of the person but no attributes of the alternatives
Further information: Multinomial logit
The utility for all alternatives depends on the same variables, sn, but the coefficients are different for different alternatives:
• Uni = βisn + εni,
• Since only differences in utility matter, it is necessary to normalize $\beta _{i}=0$ for one alternative. Assuming $\beta _{1}=0$,
• εni are iid extreme value[nb 1]
The choice probability takes the form
$P_{ni}={\exp(\beta _{i}s_{n}) \over \sum _{j=1}^{J}\exp(\beta _{j}s_{n})},$
where J is the total number of alternatives.
F. Logit with variables that vary over alternatives (also called conditional logit)
The utility for each alternative depends on attributes of that alternative, interacted perhaps with attributes of the person:
${\begin{cases}U_{ni}=\beta z_{ni}+\varepsilon _{ni}\\\varepsilon _{ni}\sim {\text{iid extreme value}}\end{cases}}\quad \Rightarrow \quad P_{ni}={\exp(\beta z_{ni}) \over \sum _{j=1}^{J}\exp(\beta z_{nj})},$
where J is the total number of alternatives.
Note that model E can be expressed in the same form as model F by appropriate respecification of variables. Define $w_{nj}^{k}=s_{n}\delta _{jk}$ where $\delta _{jk}$ is the Kronecker delta and sn are from model E. Then, model F is obtained by using
$z_{nj}=\left\{w_{nj}^{1},\cdots ,w_{nj}^{J}\right\}\quad {\text{and}}\quad \beta =\left\{\beta _{1},\cdots ,\beta _{J}\right\},$
where J is the total number of alternatives.
Multinomial choice with correlation among alternatives
A standard logit model is not always suitable, since it assumes that there is no correlation in unobserved factors over alternatives. This lack of correlation translates into a particular pattern of substitution among alternatives that might not always be realistic in a given situation. This pattern of substitution is often called the Independence of Irrelevant Alternatives (IIA) property of standard logit models. See the Red Bus/Blue Bus example in which this pattern does not hold,[12] or the path choice example.[13] A number of models have been proposed to allow correlation over alternatives and more general substitution patterns:
• Nested Logit Model - Captures correlations between alternatives by partitioning the choice set into 'nests'
• Cross-nested Logit model[14] (CNL) - Alternatives may belong to more than one nest
• C-logit Model[15] - Captures correlations between alternatives using 'commonality factor'
• Paired Combinatorial Logit Model[16] - Suitable for route choice problems.
• Generalized Extreme Value Model[17] - General class of model, derived from the random utility model[13] to which multinomial logit and nested logit belong
• Conditional probit[18][19] - Allows full covariance among alternatives using a joint normal distribution.
• Mixed logit[9][10][19]- Allows any form of correlation and substitution patterns.[20] When a mixed logit is with jointly normal random terms, the models is sometimes called "multinomial probit model with logit kernel".[13][21] Can be applied to route choice.[22]
The following sections describe Nested Logit, GEV, Probit, and Mixed Logit models in detail.
G. Nested Logit and Generalized Extreme Value (GEV) models
The model is the same as model F except that the unobserved component of utility is correlated over alternatives rather than being independent over alternatives.
• Uni = βzni + εni,
• The marginal distribution of each εni is extreme value,[nb 1] but their joint distribution allows correlation among them.
• The probability takes many forms depending on the pattern of correlation that is specified. See Generalized Extreme Value.
H. Multinomial probit
Further information: Multinomial probit
The model is the same as model G except that the unobserved terms are distributed jointly normal, which allows any pattern of correlation and heteroscedasticity:
${\begin{cases}U_{ni}=\beta z_{ni}+\varepsilon _{ni}\\\varepsilon _{n}\equiv (\varepsilon _{n1},\cdots ,\varepsilon _{nJ})\sim N(0,\Omega )\end{cases}}\quad \Rightarrow \quad P_{ni}=\Pr \left(\bigcap _{j\neq i}\beta z_{ni}+\varepsilon _{ni}>\beta z_{nj}+\varepsilon _{nj}\right)=\int I\left(\bigcap _{j\neq i}\beta z_{ni}+\varepsilon _{ni}>\beta z_{nj}+\varepsilon _{nj}\right)\phi (\varepsilon _{n}|\Omega )\;d\varepsilon _{n},$
where $\phi (\varepsilon _{n}|\Omega )$ is the joint normal density with mean zero and covariance $\Omega $.
The integral for this choice probability does not have a closed form, and so the probability is approximated by quadrature or simulation.
When $\Omega $ is the identity matrix (such that there is no correlation or heteroscedasticity), the model is called independent probit.
I. Mixed logit
Mixed Logit models have become increasingly popular in recent years for several reasons. First, the model allows $\beta $ to be random in addition to $\varepsilon $. The randomness in $\beta $ accommodates random taste variation over people and correlation across alternatives that generates flexible substitution patterns. Second, advances in simulation have made approximation of the model fairly easy. In addition, McFadden and Train have shown that any true choice model can be approximated, to any degree of accuracy by a mixed logit with appropriate specification of explanatory variables and distribution of coefficients.[20]
• Uni = βzni + εni,
• $\beta \sim f(\beta |\theta )$ for any distribution ${\it {f}}$, where $\theta $ is the set of distribution parameters (e.g. mean and variance) to be estimated,
• εni ~ iid extreme value,[nb 1]
The choice probability is
$P_{ni}=\int _{\beta }L_{ni}(\beta )f(\beta |\theta )\,d\beta ,$
where
$L_{ni}(\beta )={\exp(\beta z_{ni}) \over {\sum _{j=1}^{J}\exp(\beta z_{nj})}}$
is logit probability evaluated at $\beta ,$ with $J$ the total number of alternatives.
The integral for this choice probability does not have a closed form, so the probability is approximated by simulation.[23]
Estimation from choices
Discrete choice models are often estimated using maximum likelihood estimation. Logit models can be estimated by logistic regression, and probit models can be estimated by probit regression. Nonparametric methods, such as the maximum score estimator, have been proposed.[24][25] Estimation of such models is usually done via parametric, semi-parametric and non-parametric maximum likelihood methods,[26] but can also be done with the Partial least squares path modeling approach. [27]
Estimation from rankings
In many situations, a person's ranking of alternatives is observed, rather than just their chosen alternative. For example, a person who has bought a new car might be asked what he/she would have bought if that car was not offered, which provides information on the person's second choice in addition to their first choice. Or, in a survey, a respondent might be asked:
Example: Rank the following cell phone calling plans from your most preferred to your least preferred.
* $60 per month for unlimited anytime minutes, two-year contract with $100 early termination fee
* $30 per month for 400 anytime minutes, 3 cents per minute after 400 minutes, one-year contract with $125 early termination fee
* $35 per month for 500 anytime minutes, 3 cents per minute after 500 minutes, no contract or early termination fee
* $50 per month for 1000 anytime minutes, 5 cents per minute after 1000 minutes, two-year contract with $75 early termination fee
The models described above can be adapted to account for rankings beyond the first choice. The most prominent model for rankings data is the exploded logit and its mixed version.
J. Exploded logit
Under the same assumptions as for a standard logit (model F), the probability for a ranking of the alternatives is a product of standard logits. The model is called "exploded logit" because the choice situation that is usually represented as one logit formula for the chosen alternative is expanded ("exploded") to have a separate logit formula for each ranked alternative. The exploded logit model is the product of standard logit models with the choice set decreasing as each alternative is ranked and leaves the set of available choices in the subsequent choice.
Without loss of generality, the alternatives can be relabeled to represent the person's ranking, such that alternative 1 is the first choice, 2 the second choice, etc. The choice probability of ranking J alternatives as 1, 2, ..., J is then
$\Pr({\text{ranking }}1,2,\ldots ,J)={\exp(\beta z_{1}) \over \sum _{j=1}^{J}\exp(\beta z_{nj})}{\exp(\beta z_{2}) \over \sum _{j=2}^{J}\exp(\beta z_{nj})}\ldots {\exp(\beta z_{J-1}) \over \sum _{j=J-1}^{J}\exp(\beta z_{nj})}$
As with standard logit, the exploded logit model assumes no correlation in unobserved factors over alternatives. The exploded logit can be generalized, in the same way as the standard logit is generalized, to accommodate correlations among alternatives and random taste variation. The "mixed exploded logit" model is obtained by probability of the ranking, given above, for Lni in the mixed logit model (model I).
This model is also known in econometrics as the rank ordered logit model and it was introduced in that field by Beggs, Cardell and Hausman in 1981.[28][29] One application is the Combes et al. paper explaining the ranking of candidates to become professor.[29] It is also known as Plackett–Luce model in biomedical literature.[29][30][31]
Ordered models
In surveys, respondents are often asked to give ratings, such as:
Example: Please give your rating of how well the President is doing.
1: Very badly
2: Badly
3: Okay
4: Well
5: Very well
Or,
Example: On a 1-5 scale where 1 means disagree completely and 5 means agree completely, how much do you agree with the following statement. "The Federal government should do more to help people facing foreclosure on their homes."
A multinomial discrete-choice model can examine the responses to these questions (model G, model H, model I). However, these models are derived under the concept that the respondent obtains some utility for each possible answer and gives the answer that provides the greatest utility. It might be more natural to think that the respondent has some latent measure or index associated with the question and answers in response to how high this measure is. Ordered logit and ordered probit models are derived under this concept.
K. Ordered logit
Main article: Ordered logit
Let Un represent the strength of survey respondent n's feelings or opinion on the survey subject. Assume that there are cutoffs of the level of the opinion in choosing particular response. For instance, in the example of the helping people facing foreclosure, the person chooses
• 1, if Un < a
• 2, if a < Un < b
• 3, if b < Un < c
• 4, if c < Un < d
• 5, if Un > d,
for some real numbers a, b, c, d.
Defining $U_{n}=\beta z_{n}+\varepsilon ,\;\varepsilon \sim $ Logistic, then the probability of each possible response is:
${\begin{aligned}\Pr({\text{choosing }}1)&=\Pr(U_{n}<a)=\Pr(\varepsilon <a-\beta z_{n})={1 \over 1+\exp(-(a-\beta z_{n}))}\\\Pr({\text{choosing }}2)&=\Pr(a<U_{n}<b)=\Pr(a-\beta z_{n}<\varepsilon <b-\beta z_{n})={1 \over 1+\exp(-(b-\beta z_{n}))}-{1 \over 1+\exp(-(a-\beta z_{n}))}\\&\cdots \\\Pr({\text{choosing }}5)&=\Pr(U_{n}>d)=\Pr(\varepsilon >d-\beta z_{n})=1-{1 \over 1+\exp(-(d-\beta z_{n}))}\end{aligned}}$
The parameters of the model are the coefficients β and the cut-off points a − d, one of which must be normalized for identification. When there are only two possible responses, the ordered logit is the same a binary logit (model A), with one cut-off point normalized to zero.
L. Ordered probit
The description of the model is the same as model K, except the unobserved terms have normal distribution instead of logistic.
The choice probabilities are ($\Phi $ is the cumulative distribution function of the standard normal distribution):
${\begin{aligned}\Pr({\text{choosing }}1)&=\Phi (a-\beta z_{n})\\\Pr({\text{choosing }}2)&=\Phi (b-\beta z_{n})-\Phi (a-\beta z_{n})\\&\cdots \end{aligned}}$
See also
• Binary regression
• Dynamic discrete choice
Notes
1. The density and cumulative distribution function of the extreme value distribution are given by $f(\varepsilon _{nj})=\exp(-\varepsilon _{nj})\exp(-\exp(-\varepsilon _{nj}))$ and $F(\varepsilon _{nj})=\exp(-\exp(-\varepsilon _{nj})).$ This distribution is also called the Gumbel or type I extreme value distribution, a special type of generalized extreme value distribution.
References
1. Train, K. (1986). Qualitative Choice Analysis: Theory, Econometrics, and an Application to Automobile Demand. MIT Press. ISBN 9780262200554. Chapter 8.
2. Train, K.; McFadden, D.; Ben-Akiva, M. (1987). "The Demand for Local Telephone Service: A Fully Discrete Model of Residential Call Patterns and Service Choice". RAND Journal of Economics. 18 (1): 109–123. doi:10.2307/2555538. JSTOR 2555538.
3. Train, K.; Winston, C. (2007). "Vehicle Choice Behavior and the Declining Market Share of US Automakers". International Economic Review. 48 (4): 1469–1496. doi:10.1111/j.1468-2354.2007.00471.x. S2CID 13085087.
4. Fuller, W. C.; Manski, C.; Wise, D. (1982). "New Evidence on the Economic Determinants of Post-secondary Schooling Choices". Journal of Human Resources. 17 (4): 477–498. doi:10.2307/145612. JSTOR 145612.
5. Train, K. (1978). "A Validation Test of a Disaggregate Mode Choice Model" (PDF). Transportation Research. 12 (3): 167–174. doi:10.1016/0041-1647(78)90120-x.
6. Baltas, George; Doyle, Peter (2001). "Random utility models in marketing research: a survey". Journal of Business Research. 51 (2): 115–125. doi:10.1016/S0148-2963(99)00058-2.
7. Ramming, M. S. (2001). Network Knowledge and Route Choice (Thesis). Unpublished Ph.D. Thesis, Massachusetts Institute of Technology. MIT catalogue. hdl:1721.1/49797.
8. Goett, Andrew; Hudson, Kathleen; Train, Kenneth E. (2002). "Customer Choice Among Retail Energy Suppliers". Energy Journal. 21 (4): 1–28.
9. Revelt, David; Train, Kenneth E. (1998). "Mixed Logit with Repeated Choices: Households' Choices of Appliance Efficiency Level". Review of Economics and Statistics. 80 (4): 647–657. doi:10.1162/003465398557735. JSTOR 2646846. S2CID 10423121.
10. Train, Kenneth E. (1998). "Recreation Demand Models with Taste Variation". Land Economics. 74 (2): 230–239. CiteSeerX 10.1.1.27.4879. doi:10.2307/3147053. JSTOR 3147053.
11. Cooper, A. B.; Millspaugh, J. J. (1999). "The application of discrete choice models to wildlife resource selection studies". Ecology. 80 (2): 566–575. doi:10.1890/0012-9658(1999)080[0566:TAODCM]2.0.CO;2.
12. Ben-Akiva, M.; Lerman, S. (1985). Discrete Choice Analysis: Theory and Application to Travel Demand. Transportation Studies. Massachusetts: MIT Press.
13. Ben-Akiva, M.; Bierlaire, M. (1999). "Discrete Choice Methods and Their Applications to Short Term Travel Decisions" (PDF). In Hall, R. W. (ed.). Handbook of Transportation Science.
14. Vovsha, P. (1997). "Application of Cross-Nested Logit Model to Mode Choice in Tel Aviv, Israel, Metropolitan Area". Transportation Research Record. 1607: 6–15. doi:10.3141/1607-02. S2CID 110401901. Archived from the original on 2013-01-29.
15. Cascetta, E.; Nuzzolo, A.; Russo, F.; Vitetta, A. (1996). "A Modified Logit Route Choice Model Overcoming Path Overlapping Problems: Specification and Some Calibration Results for Interurban Networks" (PDF). In Lesort, J. B. (ed.). Transportation and Traffic Theory. Proceedings from the Thirteenth International Symposium on Transportation and Traffic Theory. Lyon, France: Pergamon. pp. 697–711.
16. Chu, C. (1989). "A Paired Combinatorial Logit Model for Travel Demand Analysis". Proceedings of the 5th World Conference on Transportation Research. Vol. 4. Ventura, CA. pp. 295–309.{{cite book}}: CS1 maint: location missing publisher (link)
17. McFadden, D. (1978). "Modeling the Choice of Residential Location" (PDF). In Karlqvist, A.; et al. (eds.). Spatial Interaction Theory and Residential Location. Amsterdam: North Holland. pp. 75–96.
18. Hausman, J.; Wise, D. (1978). "A Conditional Probit Model for Qualitative Choice: Discrete Decisions Recognizing Interdependence and Heterogenous Preferences". Econometrica. 48 (2): 403–426. doi:10.2307/1913909. JSTOR 1913909.
19. Train, K. (2003). Discrete Choice Methods with Simulation. Massachusetts: Cambridge University Press.
20. McFadden, D.; Train, K. (2000). "Mixed MNL Models for Discrete Response" (PDF). Journal of Applied Econometrics. 15 (5): 447–470. CiteSeerX 10.1.1.68.2871. doi:10.1002/1099-1255(200009/10)15:5<447::AID-JAE570>3.0.CO;2-1.
21. Ben-Akiva, M.; Bolduc, D. (1996). "Multinomial Probit with a Logit Kernel and a General Parametric Specification of the Covariance Structure" (PDF). Working Paper.
22. Bekhor, S.; Ben-Akiva, M.; Ramming, M. S. (2002). "Adaptation of Logit Kernel to Route Choice Situation". Transportation Research Record. 1805: 78–85. doi:10.3141/1805-10. S2CID 110895210. Archived from the original on 2012-07-17.
23. . Also see Mixed logit for further details.
24. Manski, Charles F. (1975). "Maximum score estimation of the stochastic utility model of choice". Journal of Econometrics. Elsevier BV. 3 (3): 205–228. doi:10.1016/0304-4076(75)90032-9. ISSN 0304-4076.
25. Horowitz, Joel L. (1992). "A Smoothed Maximum Score Estimator for the Binary Response Model". Econometrica. JSTOR. 60 (3): 505–531. doi:10.2307/2951582. ISSN 0012-9682. JSTOR 2951582.
26. Park, Byeong U.; Simar, Léopold; Zelenyuk, Valentin (2017). "Nonparametric estimation of dynamic discrete choice models for time series data" (PDF). Computational Statistics & Data Analysis. 108: 97–120. doi:10.1016/j.csda.2016.10.024.
27. Hair, J.F.; Ringle, C.M.; Gudergan, S.P.; Fischer, A.; Nitzl, C.; Menictas, C. (2019). "Partial least squares structural equation modeling-based discrete choice modeling: an illustration in modeling retailer choice" (PDF). Business Research. 12: 115–142. doi:10.1007/s40685-018-0072-4.
28. Beggs, S.; Cardell, S.; Hausman, J. (1981). "Assessing the Potential Demand for Electric Cars". Journal of Econometrics. 17 (1): 1–19. doi:10.1016/0304-4076(81)90056-7.
29. Combes, Pierre-Philippe; Linnemer, Laurent; Visser, Michael (2008). "Publish or Peer-Rich? The Role of Skills and Networks in Hiring Economics Professors". Labour Economics. 15 (3): 423–441. doi:10.1016/j.labeco.2007.04.003.
30. Plackett, R. L. (1975). "The Analysis of Permutations". Journal of the Royal Statistical Society, Series C. 24 (2): 193–202. doi:10.2307/2346567. JSTOR 2346567.
31. Luce, R. D. (1959). Individual Choice Behavior: A Theoretical Analysis. Wiley.
Further reading
• Anderson, S., A. de Palma and J.-F. Thisse (1992), Discrete Choice Theory of Product Differentiation, MIT Press,
• Ben-Akiva, M.; Lerman, S. (1985). Discrete Choice Analysis: Theory and Application to Travel Demand. MIT Press.
• Greene, William H. (2012). Econometric Analysis (Seventh ed.). Upper Saddle River: Pearson Prentice-Hall. pp. 770–862. ISBN 978-0-13-600383-0.
• Hensher, D.; Rose, J.; Greene, W. (2005). Applied Choice Analysis: A Primer. Cambridge University Press.
• Maddala, G. (1983). Limited-dependent and Qualitative Variables in Econometrics. Cambridge University Press.
• McFadden, Daniel L. (1984). Econometric analysis of qualitative response models. Handbook of Econometrics, Volume II. Vol. Chapter 24. Elsevier Science Publishers BV.
• Train, K. (2009) [2003]. Discrete Choice Methods with Simulation. Cambridge University Press.
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Wikipedia
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Qualitative risk analysis
Qualitative risk analysis is a technique used to quantify risk associated with a particular hazard. Risk assessment is used for uncertain events that could have many outcomes and for which there could be significant consequences. Risk is a function of probability of an event (a particular hazard occurring) and the consequences given the event occurs. Probability refers to the likelihood that a hazard will occur. In a qualitative assessment, probability and consequence are not numerically estimated, but are evaluated verbally using qualifiers like high likelihood, low likelihood, etc. Qualitative assessments are good for screening level assessments when comparing/screening multiple alternatives or for when sufficient data is not available to support numerical probability or consequence estimates. Once numbers are inserted into the analysis (either by quantifying the likelihood of a hazard or quantifying the consequences) the analysis transitions to a semi-quantitative or quantitative risk assessment.
Qualitative techniques
There are several techniques when performing qualitative risk analysis to determine the probability and impact of risks, including the following:
• Brainstorming, interviewing, Delphi technique
• Historical data
• Strength, weakness, opportunity, and threats analysis (SWOT analysis)
• Risk rating scales
Developing rating scales
Assigning probability and impacts to risks is a subjective exercise. Some of this subjectivity can be eliminated by developing rating scales that are agreed upon by the sponsor, project manager, and key team members. Some organizations, particularly those that have project management offices responsible for overseeing all projects, have rating scales already developed.[1]
External links
• Current Intelligence Bulletin 69: NIOSH Practices in Occupational Risk Assessment. National Institute for Occupational Safety and Health (NIOSH), USA.
References
1. Kim Heldman, PMP, Project Manager's Spotlight on Risk Management,p125,126
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Qualitative theory of differential equations
In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relatively few differential equations that can be solved explicitly, but using tools from analysis and topology, one can "solve" them in the qualitative sense, obtaining information about their properties.[1]
References
1. "Qualitative theory of differential equations", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Further reading
• Viktor Vladimirovich Nemytskii, Vyacheslav Stepanov, Qualitative theory of differential equations, Princeton University Press, Princeton, 1960.
Original references
• Henri Poincaré, "Mémoire sur les courbes définies par une équation différentielle", Journal de Mathématiques Pures et Appliquées (1881, in French)
• Lyapunov, Aleksandr M. (1992). "The general problem of the stability of motion". International Journal of Control. 55 (3): 531–534. doi:10.1080/00207179208934253. ISSN 0020-7179. (it was translated from the original Russian into French and then into this English version, the original is from the year 1892)
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Quanta Magazine
Quanta Magazine is an editorially independent[1] online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science.
Quanta Magazine
EditorThomas Lin
CategoriesPhysics, mathematics, biology, computer science
PublisherSimons Foundation
First issue2012
CountryUnited States
Websitewww.quantamagazine.org
ISSN2640-2661
OCLC914339324
Undark Magazine described Quanta Magazine as "highly regarded for its masterful coverage of complex topics in science and math."[2] The science news aggregator RealClearScience ranked Quanta Magazine first on its list of "The Top 10 Websites for Science in 2018."[3] In 2020, the magazine received a National Magazine Award for General Excellence from the American Society of Magazine Editors for its "willingness to tackle some of the toughest and most difficult topics in science and math in a language that is accessible to the lay reader without condescension or oversimplification."
The articles in the magazine are freely available to read online.[4] Scientific American,[5] Wired,[6] The Atlantic, and The Washington Post, as well as international science publications like Spektrum der Wissenschaft,[7] have reprinted articles from the magazine.
History
Quanta Magazine was initially launched as Simons Science News[8] in October 2012, but it was renamed to its current title in July 2013.[9] It was founded by the former New York Times journalist Thomas Lin, who is the magazine's editor-in-chief.[10][11] The two deputy editors are John Rennie and Michael Moyer, formerly of Scientific American, and the art director is Samuel Velasco.
In November 2018, MIT Press published two collections of articles from Quanta Magazine, Alice and Bob Meet the Wall of Fire[12] and The Prime Number Conspiracy.[13]
In May 2022 the magazine's staff, notably Natalie Wolchover, were awarded the Pulitzer Prize for Explanatory Reporting.[14]
References
1. "About Quanta Magazine". Quanta Magazine. Simons Foundation. Retrieved 6 November 2019.
2. Robin Lloyd (5 April 2017). "Hard-Sciences Magazine Goes to the Next Level". Undark Magazine. Retrieved 6 November 2019.
3. Ross Pomeroy (2018-12-10). "The Top 10 Websites for Science in 2018". RealClearScience. Retrieved 6 November 2019.
4. Richard Elwes (6 November 2013). "Quanta Magazine". London Mathematical Society. Retrieved 6 November 2019.
5. "Stories by Quanta Magazine". Scientific American. Retrieved 6 November 2019.
6. "Quanta Magazine". Wired. Retrieved 6 November 2019.
7. "Quanta Magazine". Spektrum der Wissenschaft. Retrieved 6 November 2019.
8. Dennis Overbye (6 May 2013). "A Magazine or a Living Fossil?". The New York Times. Retrieved 6 November 2019.
9. Carl Zimmer. "How Things Get Complex: My New Story for Scientific American & Quanta Magazine". National Geographic. Retrieved 6 November 2019.
10. Jonathan Wai (16 June 2014). "Reinventing The Boundaries of Science Journalism". Psychology Today. Retrieved 6 November 2019.
11. Kara Bloomgarden-Smoke (20 May 2016). "Quanta Magazine's Thomas Lin Spends His Days 'Illuminating Science'". Observer. Retrieved 6 November 2019.
12. Thomas Lin, ed. (2018). Alice and Bob Meet the Wall of Fire: The Biggest Ideas in Science from Quanta [sic]. Cambridge, Massachusetts: MIT Press. ISBN 9780262536349.
13. Thomas Lin, ed. (2018). The Prime Number Conspiracy: The Biggest Ideas in Math from Quanta [sic]. Cambridge, Massachusetts: MIT Press. ISBN 9780262536356.
14. "The 2022 Pulitzer Prize Announcement". pulitzer.org.
External links
• Official website
• QuantaScienceChannel
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Quantaloid
In mathematics, a quantaloid is a category enriched over the category Sup of suplattices.[1] In other words, for any objects a and b the morphism object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins:
$(\bigvee _{i}f_{i})\circ (\bigvee _{j}g_{j})=\bigvee _{i,j}(f_{i}\circ g_{j})$
The endomorphism lattice $\mathrm {Hom} (X,X)$ of any object $X$ in a quantaloid is a quantale, whence the name.
References
1. Rosenthal, Kimmo I. (1996), The theory of quantaloids, Pitman Research Notes in Mathematics Series, vol. 348, Longman, Harlow, ISBN 0-582-29440-1, MR 1427263. See in particular p. 15.
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Quantification (machine learning)
In machine learning and data mining, quantification (variously called learning to quantify, or supervised prevalence estimation, or class prior estimation) is the task of using supervised learning in order to train models (quantifiers) that estimate the relative frequencies (also known as prevalence values) of the classes of interest in a sample of unlabelled data items.[1][2] For instance, in a sample of 100,000 unlabelled tweets known to express opinions about a certain political candidate, a quantifier may be used to estimate the percentage of these 100,000 tweets which belong to class `Positive' (i.e., which manifest a positive stance towards this candidate), and to do the same for classes `Neutral' and `Negative'.[3]
Quantification may also be viewed as the task of training predictors that estimate a (discrete) probability distribution, i.e., that generate a predicted distribution that approximates the unknown true distribution of the items across the classes of interest. Quantification is different from classification, since the goal of classification is to predict the class labels of individual data items, while the goal of quantification it to predict the class prevalence values of sets of data items. Quantification is also different from regression, since in regression the training data items have real-valued labels, while in quantification the training data items have class labels.
It has been shown in multiple research works[4][5][6][7][8] that performing quantification by classifying all unlabelled instances and then counting the instances that have been attributed to each class (the 'classify and count' method) usually leads to suboptimal quantification accuracy. This suboptimality may be seen as a direct consequence of 'Vapnik's principle', which states:
If you possess a restricted amount of information for solving some problem, try to solve the problem directly and never solve a more general problem as an intermediate step. It is possible that the available information is sufficient for a direct solution but is insufficient for solving a more general intermediate problem.[9]
In our case, the problem to be solved directly is quantification, while the more general intermediate problem is classification. As a result of the suboptimality of the 'classify and count' method, quantification has evolved as a task in its own right, different (in goals, methods, techniques, and evaluation measures) from classification.
Quantification tasks
The main variants of quantification, according to the characteristics of the set of classes used, are:
• Binary quantification, corresponding to the case in which there are only $n=2$ classes and each data item belongs to exactly one of them;
• Single-label multiclass quantification, corresponding to the case with $n>2$ classes and each data item belongs to exactly one of them;[10]
• Ordinal quantification, corresponding to the single-label multiclass case in which a total order is defined on the set of classes.
Most known quantification methods address the binary case or the single-label multiclass case, and only few of them address the ordinal case.
Binary-only methods include the Mixture Model (MM) method,[4] the HDy method,[11] SVM(KLD),[7] and SVM(Q).[6]
Methods that can deal with both the binary case and the single-label multiclass case include probabilistic classify and count (PCC),[5] adjusted classify and count (ACC),[4] probabilistic adjusted classify and count (PACC),[5] and the Saerens-Latinne-Decaestecker EM-based method (SLD).[12]
Methods for the ordinal case include Ordinal Quantification Tree (OQT),[13] and ordinal version of the above-mentioned ACC, PACC, and SLD methods.[14]
Evaluation measures for quantification
Several evaluation measures can be used for evaluating the error of a quantification method. Since quantification consists of generating a predicted probability distribution that estimates a true probability distribution, these evaluation measures are ones that compare two probability distributions. Most evaluation measures for quantification belong to the class of divergences. Evaluation measures for binary quantification and single-label multiclass quantification are[15]
• Absolute Error
• Squared Error
• Relative Absolute Error
• Kullback-Leibler divergence
• Pearson Divergence
Evaluation measures for ordinal quantification are
• Normalized Match Distance (a particular case of the Earth Mover's Distance)
• Root Normalized Order-Aware Distance
Applications
Quantification is of special interest in fields such as the social sciences,[16] epidemiology,[17] market research, and ecological modelling,[18] since these fields are inherently concerned with aggregate data; however, quantification is also useful in applications outside these fields, such as in measuring classifier bias[19] and enforcing classifier fairness,[20] performing word sense disambiguation,[21] allocating resources,[4] and improving the accuracy of classifiers.[12]
Resources
• LQ 2021: the 1st International Workshop on Learning to Quantify[22]
• LQ 2022: the 2nd International Workshop on Learning to Quantify[23]
• LeQua 2022: A machine learning competition on Learning to Quantify [24]
• QuaPy: An open-source Python-based software library for quantification[25]
References
1. Pablo González; Alberto Castaño; Nitesh Chawla; Juan José del Coz (2017). "A review on quantification learning". ACM Computing Surveys. 50 (5): 74:1–74:40. doi:10.1145/3117807. hdl:10651/45313. S2CID 38185871.
2. Andrea Esuli; Alessandro Fabris; Alejandro Moreo; Fabrizio Sebastiani (2023). Learning to Quantify. The Information Retrieval Series. Vol. 47. Cham, CH: Springer Nature. doi:10.1007/978-3-031-20467-8. ISBN 978-3-031-20466-1. S2CID 257560090.
3. Hopkins, Daniel J.; King, Gary (2010). "A Method of Automated Nonparametric Content Analysis for Social Science". American Journal of Political Science. 54 (1): 229–247. doi:10.1111/j.1540-5907.2009.00428.x. ISSN 0092-5853. JSTOR 20647981. S2CID 1177676.
4. George Forman (2008). "Quantifying counts and costs via classification". Data Mining and Knowledge Discovery. 17 (2): 164–206. doi:10.1007/s10618-008-0097-y. S2CID 1435935.
5. Antonio Bella; Cèsar Ferri; José Hernández-Orallo; María José Ramírez-Quintana (2010). "Quantification via Probability Estimators". 2010 IEEE International Conference on Data Mining. pp. 737–742. doi:10.1109/icdm.2010.75. ISBN 978-1-4244-9131-5. S2CID 9670485.
6. José Barranquero; Jorge Díez; Juan José del Coz (2015). "Quantification-oriented learning based on reliable classifiers". Pattern Recognition. 48 (2): 591–604. Bibcode:2015PatRe..48..591B. doi:10.1016/j.patcog.2014.07.032. hdl:10651/30611.
7. Andrea Esuli; Fabrizio Sebastiani (2015). "Optimizing text quantifiers for multivariate loss functions". ACM Transactions on Knowledge Discovery from Data. 9 (4): Article 27. arXiv:1502.05491. doi:10.1145/2700406. S2CID 16824608.
8. Wei Gao; Fabrizio Sebastiani (2016). "From classification to quantification in tweet sentiment analysis". Social Network Analysis and Mining. 6 (19): 1–22. doi:10.1007/s13278-016-0327-z. S2CID 15631612.
9. Vladimir Vapnik (1998). Statistical learning theory. New York, US: Wiley.
10. Jerzak, Connor T.; King, Gary; Strezhnev, Anton (2022). "An Improved Method of Automated Nonparametric Content Analysis for Social Science". Political Analysis. 31 (1): 42–58. doi:10.1017/pan.2021.36. ISSN 1047-1987. S2CID 3796379.
11. Víctor González-Castro; Rocío Alaiz-Rodríguez; Enrique Alegre (2013). "Class distribution estimation based on the {H}ellinger distance". Information Sciences. 218: 146–164. doi:10.1016/j.ins.2012.05.028.
12. Marco Saerens; Patrice Latinne; Christine Decaestecker (2002). "Adjusting the outputs of a classifier to new a priori probabilities: A simple procedure" (PDF). Neural Computation. 14 (1): 21–41. doi:10.1162/089976602753284446. PMID 11747533. S2CID 18254013.
13. Giovanni Da San Martino; Wei Gao; Fabrizio Sebastiani (2016). "Ordinal Text Quantification". Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval. pp. 937–940. doi:10.1145/2911451.2914749. ISBN 9781450340694. S2CID 8102324.{{cite book}}: CS1 maint: date and year (link)
14. Mirko Bunse; Alejandro Moreo; Fabrizio Sebastiani; Marin Senz (2022). "Ordinal quantification through regularization". Proceedings of the 33rd European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML / PKDD 2022), Grenoble, FR.
15. Fabrizio Sebastiani (2020). "Evaluation measures for quantification: An axiomatic approach". Information Retrieval Journal. 23 (3): 255–288. arXiv:1809.01991. doi:10.1007/s10791-019-09363-y. S2CID 52170301.
16. Daniel J. Hopkins; Gary King (2010). "A method of automated nonparametric content analysis for social science". American Journal of Political Science. 54 (1): 229–247. doi:10.1111/j.1540-5907.2009.00428.x. S2CID 1177676.
17. Gary King; Ying Lu (2008). "Verbal autopsy methods with multiple causes of death". Statistical Science. 23 (1): 78–91. arXiv:0808.0645. doi:10.1214/07-sts247. S2CID 4084198.
18. Pablo González; Eva Álvarez; Jorge Díez; Ángel López-Urrutia; Juan J. del Coz (2017). "Validation methods for plankton image classification systems" (PDF). Limnology and Oceanography: Methods. 15 (3): 221–237. doi:10.1002/lom3.10151. S2CID 59438870.
19. Alessandro Fabris; Andrea Esuli; Alejandro Moreo; Fabrizio Sebastiani (2023). "Measuring Fairness Under Unawareness of Sensitive Attributes: A Quantification-Based Approach". Journal of Artificial Intelligence Research. 76: 1117–1180. arXiv:2109.08549. doi:10.1613/jair.1.14033. S2CID 247315416.
20. Arpita Biswas; Suvam Mukherjee (2019). "Fairness through the lens of proportional equality". Proceedings of the 18th International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS 2019). Montreal, CA: 1832–1834. ISBN 9781450363099.
21. Yee Seng Chan; Hwee Tou Ng (2005). "Word sense disambiguation with distribution estimation". Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005). Edinburgh, UK: 1010–1015.
22. "LQ 2021: the 1st International Workshop on Learning to Quantify".
23. "LQ 2022: the 2nd International Workshop on Learning to Quantify".
24. "LeQua 2022: A machine learning competition on Learning to Quantify".
25. "QuaPy: A Python-Based Framework for Quantification". GitHub. 23 November 2021.
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Wikipedia
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Quantification
Quantification may refer to:
• Quantification (science), the act of counting and measuring
• Quantification (machine learning), the task of estimating class prevalence values in unlabelled data
• Quantifier (linguistics), an indicator of quantity
• Quantifier (logic)
Look up quantification in Wiktionary, the free dictionary.
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Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "$\exists x$ such that $\ldots $" can be viewed as a question "When is there an $x$ such that $\ldots $?", and the statement without quantifiers can be viewed as the answer to that question.[1]
One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula $\alpha $, there exists another formula $\alpha _{QF}$ without quantifiers that is equivalent to it (modulo this theory).
Examples
An example from high school mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative:[1]
$\exists x\in \mathbb {R} .(a\neq 0\wedge ax^{2}+bx+c=0)\ \ \Longleftrightarrow \ \ a\neq 0\wedge b^{2}-4ac\geq 0$
Here the sentence on the left-hand side involves a quantifier $\exists x\in \mathbb {R} $, while the equivalent sentence on the right does not.
Examples of theories that have been shown decidable using quantifier elimination are Presburger arithmetic,[2][3][4][5][6] algebraically closed fields, real closed fields,[6] [7] atomless Boolean algebras, term algebras, dense linear orders,[6] abelian groups,[8] random graphs, as well as many of their combinations such as Boolean algebra with Presburger arithmetic, and term algebras with queues.
Quantifier eliminator for the theory of the real numbers as an ordered additive group is Fourier–Motzkin elimination; for the theory of the field of real numbers it is the Tarski–Seidenberg theorem.[6]
Quantifier elimination can also be used to show that "combining" decidable theories leads to new decidable theories (see Feferman-Vaught theorem).
Algorithms and decidability
If a theory has quantifier elimination, then a specific question can be addressed: Is there a method of determining $\alpha _{QF}$ for each $\alpha $? If there is such a method we call it a quantifier elimination algorithm. If there is such an algorithm, then decidability for the theory reduces to deciding the truth of the quantifier-free sentences. Quantifier-free sentences have no variables, so their validity in a given theory can often be computed, which enables the use of quantifier elimination algorithms to decide validity of sentences.
Related concepts
Various model-theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
Every first-order theory with quantifier elimination is model complete. Conversely, a model-complete theory, whose theory of universal consequences has the amalgamation property, has quantifier elimination.[9]
The models of the theory of the universal consequences of a theory $T$ are precisely the substructures of the models of $T$.[9] The theory of linear orders does not have quantifier elimination. However the theory of its universal consequences has the amalgamation property.
Basic ideas
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form:
$\exists x.\bigwedge _{i=1}^{n}L_{i}$
where each $L_{i}$ is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of literals, then if $F$ is a quantifier-free formula, we can write it in disjunctive normal form
$\bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij},$
and use the fact that
$\exists x.\bigvee _{j=1}^{m}\bigwedge _{i=1}^{n}L_{ij}$
is equivalent to
$\bigvee _{j=1}^{m}\exists x.\bigwedge _{i=1}^{n}L_{ij}.$
Finally, to eliminate a universal quantifier
$\forall x.F$
where $F$ is quantifier-free, we transform $\lnot F$ into disjunctive normal form, and use the fact that $\forall x.F$ is equivalent to $\lnot \exists x.\lnot F.$
Relationship with decidability
In early model theory, quantifier elimination was used to demonstrate that various theories possess properties like decidability and completeness. A common technique was to show first that a theory admits elimination of quantifiers and thereafter prove decidability or completeness by considering only the quantifier-free formulas. This technique can be used to show that Presburger arithmetic is decidable.
Theories could be decidable yet not admit quantifier elimination. Strictly speaking, the theory of the additive natural numbers did not admit quantifier elimination, but it was an expansion of the additive natural numbers that was shown to be decidable. Whenever a theory is decidable, and the language of its valid formulas is countable, it is possible to extend the theory with countably many relations to have quantifier elimination (for example, one can introduce, for each formula of the theory, a relation symbol that relates the free variables of the formula).[10]
Example: Nullstellensatz for algebraically closed fields and for differentially closed fields.
See also
• Cylindrical algebraic decomposition
• Elimination theory
• Conjunction elimination
Notes
1. Brown 2002.
2. Presburger 1929.
3. Monk 2012, p. 240.
4. Nipkow 2010.
5. Enderton 2001, p. 188.
6. Grädel et al. 2007.
7. Fried & Jarden 2008, p. 171.
8. Szmielew 1955, Page 229 describes "the method of eliminating quantification.".
9. Hodges 1993.
10. "Proofs with Quantifiers 2 | Stanford University - KeepNotes". keepnotes.com. Retrieved 2023-08-10.
References
• Brown, Christopher W. (July 31, 2002). "What is Quantifier Elimination". Retrieved Aug 30, 2018.
• Enderton, Herbert (2001). A mathematical introduction to logic (2nd ed.). Boston, MA: Academic Press. ISBN 978-0-12-238452-3.
• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.
• Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007). Finite model theory and its applications. Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer-Verlag. ISBN 978-3-540-00428-8. Zbl 1133.03001.
• Hodges, Wilfrid (1993). Model Theory. Encyclopedia of Mathematics and its Applications. Vol. 42. Cambridge University Press. doi:10.1017/CBO9780511551574. ISBN 9780521304429.
• Kuncak, Viktor; Rinard, Martin (2003). "Structural subtyping of non-recursive types is decidable" (PDF). 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings. pp. 96–107. doi:10.1109/LICS.2003.1210049. ISBN 0-7695-1884-2. S2CID 14182674.
• Monk, J. Donald Monk (2012). Mathematical Logic (Graduate Texts in Mathematics (37)) (Softcover reprint of the original 1st ed. 1976 ed.). Springer. ISBN 9781468494549.
• Nipkow, T (2010). "Linear Quantifier Elimination" (PDF). Journal of Automated Reasoning. 45 (2): 189–212. doi:10.1007/s10817-010-9183-0. S2CID 14279141. Retrieved 2022-11-12.
• Presburger, Mojżesz (1929). "Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt". Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, Warszawa: 92–101., see Stansifer (1984) for an English translation
• Stansifer, Ryan (Sep 1984). Presburger's Article on Integer Arithmetic: Remarks and Translation (PDF) (Technical Report). Vol. TR84-639. Ithaca, New York: Dept. of Computer Science, Cornell University.
• Szmielew, Wanda (1955). "Elementary properties of Abelian groups". Fundamenta Mathematicae. 41 (2): 203–271. doi:10.4064/fm-41-2-203-271. MR 0072131.
• Jeannerod, Nicolas; Treinen, Ralf. Deciding the First-Order Theory of an Algebra of Feature Trees with Updates. International Joint Conference on Automated Reasoning (IJCAR). doi:10.1007/978-3-319-94205-6_29.
• Sturm, Thomas (2017). "A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications". Mathematics in Computer Science. 11 (3–4): 483–502. doi:10.1007/s11786-017-0319-z.
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Wikipedia
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Quantificational variability effect
Quantificational variability effect (QVE) is the intuitive equivalence of certain sentences with quantificational adverbs (Q-adverbs) and sentences without these, but with quantificational determiner phrases (DP) in argument position instead.
• 1. (a) A cat is usually smart. (Q-adverb)
• 1. (b) Most cats are smart. (DP)
• 2. (a) A dog is always smart. (Q-adverb)
• 2. (b) All dogs are smart. (DP)[1]
Analysis of QVE is widely cited as entering the literature with David Lewis' "Adverbs of Quantification" (1975), where he proposes QVE as a solution to Peter Geach's donkey sentence (1962). Terminology, and comprehensive analysis, is normally attributed to Stephen Berman's "Situation-Based Semantics for Adverbs of Quantification" (1987).
See also
• David Kellogg Lewis
• Donkey pronoun
• Existential closure
• Irene Heim
Notes
1. Adapted from Endriss and Hinterwimmer (2005).
Literature
Core texts
• Berman, Stephen. The Semantics of Open Sentences. PhD thesis. University of Massachusetts Amherst, 1991.
• Berman, Stephen. 'An Analysis of Quantifier Variability in Indirect Questions'. In MIT Working Papers in Linguistics 11. Edited by Phil Branigan and others. Cambridge: MIT Press, 1989. Pages 1–16.
• Berman, Stephen. 'Situation-Based Semantics for Adverbs of Quantification'. In University of Massachusetts Occasional Papers 12. Edited by J. Blevins and Anne Vainikka. Graduate Linguistic Student Association (GLSA), University of Massachusetts Amherst, 1987. Pages 45–68.
Select bibliography
External links
Core text
• Lewis, David. 'Adverbs of Quantification'. In Formal Semantics of Natural Language. Edited by Edward L Keenan. Cambridge: Cambridge University Press, 1975. Pages 3–15.
Other texts available online
• Endriss, Cornelia and Stefan Hinterwimmer. 'The Non-Uniformity of Quantificational Variability Effects: A Comparison of Singular Indefinites, Bare Plurals and Plural Definites'. Belgian Journal of Linguistics 19 (2005): 93–120.
Formal semantics (natural language)
Central concepts
• Compositionality
• Denotation
• Entailment
• Extension
• Generalized quantifier
• Intension
• Logical form
• Presupposition
• Proposition
• Reference
• Scope
• Speech act
• Syntax–semantics interface
• Truth conditions
Topics
Areas
• Anaphora
• Ambiguity
• Binding
• Conditionals
• Definiteness
• Disjunction
• Evidentiality
• Focus
• Indexicality
• Lexical semantics
• Modality
• Negation
• Propositional attitudes
• Tense–aspect–mood
• Quantification
• Vagueness
Phenomena
• Antecedent-contained deletion
• Cataphora
• Coercion
• Conservativity
• Counterfactuals
• Cumulativity
• De dicto and de re
• De se
• Deontic modality
• Discourse relations
• Donkey anaphora
• Epistemic modality
• Exhaustivity
• Faultless disagreement
• Free choice inferences
• Givenness
• Crossover effects
• Hurford disjunction
• Inalienable possession
• Intersective modification
• Logophoricity
• Mirativity
• Modal subordination
• Opaque contexts
• Performatives
• Polarity items
• Privative adjectives
• Quantificational variability effect
• Responsive predicate
• Rising declaratives
• Scalar implicature
• Sloppy identity
• Subsective modification
• Subtrigging
• Telicity
• Temperature paradox
• Veridicality
Formalism
Formal systems
• Alternative semantics
• Categorial grammar
• Combinatory categorial grammar
• Discourse representation theory (DRT)
• Dynamic semantics
• Frame semantics
• Generative grammar
• Glue semantics
• Inquisitive semantics
• Intensional logic
• Lambda calculus
• Mereology
• Montague grammar
• Segmented discourse representation theory (SDRT)
• Situation semantics
• Supervaluationism
• Type theory
• TTR
Concepts
• Autonomy of syntax
• Context set
• Continuation
• Conversational scoreboard
• Existential closure
• Function application
• Meaning postulate
• Monads
• Possible world
• Quantifier raising
• Quantization
• Question under discussion
• Semantic parsing
• Squiggle operator
• Strict conditional
• Type shifter
• Universal grinder
See also
• Cognitive semantics
• Computational semantics
• Distributional semantics
• Formal grammar
• Inferentialism
• Term logic
• Linguistics wars
• Philosophy of language
• Pragmatics
• Context
• Deixis
• Semantics of logic
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Wikipedia
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Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier $\forall $ in the first order formula $\forall xP(x)$ expresses that everything in the domain satisfies the property denoted by $P$. On the other hand, the existential quantifier $\exists $ in the formula $\exists xP(x)$ expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.
The most commonly used quantifiers are $\forall $ and $\exists $. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula $\neg \exists xP(x)$ which expresses that nothing has the property $P$. Other quantifiers are only definable within second order logic or higher order logics. Quantifiers have been generalized beginning with the work of Mostowski and Lindström.
In a first-order logic statement, quantifications in the same type (either universal quantifications or existential quantifications) can be exchanged without changing the meaning of the statement, while the exchange of quantifications in different types changes the meaning. As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications.
First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of generalized quantifiers.
Relations to logical conjunction and disjunction
For a finite domain of discourse $D=\{a_{1},...a_{n}\}$, the universally quantified formula $\forall x\in D\;P(x)$ is equivalent to the logical conjunction $P(a_{1})\land ...\land P(a_{n})$. Dually, the existentially quantified formula $\exists x\in D\;P(x)$ is equivalent to the logical disjunction $P(a_{1})\lor ...\lor P(a_{n})$. For example, if $B=\{0,1\}$ is the set of binary digits, the formula $\forall x\in B\;x=x^{2}$ abbreviates $0=0^{2}\land 1=1^{2}$, which evaluates to true.
Infinite domain of discourse
Consider the following statement (using dot notation for multiplication):
1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., etc.
This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages, this is immediately a problem, since syntax rules are expected to generate finite words.
The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if an assertion were to be made about every irrational number, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct, equivalent formulation which avoids these problems uses universal quantification:
For each natural number n, n · 2 = n + n.
A similar analysis applies to the disjunction,
1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc.
which can be rephrased using existential quantification:
For some natural number n, n is equal to 5+5.
Algebraic approaches to quantification
It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow and interest in such algebra has been limited. Three approaches have been devised to date:
• Relation algebra, invented by Augustus De Morgan, and developed by Charles Sanders Peirce, Ernst Schröder, Alfred Tarski, and Tarski's students. Relation algebra cannot represent any formula with quantifiers nested more than three deep. Surprisingly, the models of relation algebra include the axiomatic set theory ZFC and Peano arithmetic;
• Cylindric algebra, devised by Alfred Tarski, Leon Henkin, and others;
• The polyadic algebra of Paul Halmos.
Notation
The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is "∀", a rotated letter "A", which stands for "for all" or "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for "there exists" or "exists".[2][3]
An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let X be the set of all Peter's friends, P(x) the predicate "x likes to dance", and Q(x) the predicate "x likes to go to the beach". Then the above sentence can be written in formal notation as $\forall {x}{\in }X,(P(x)\lor Q(x))$, which is read, "for every x that is a member of X, P applies to x or Q applies to x".
Some other quantified expressions are constructed as follows,
$\exists {x}\,P$[4] $\forall {x}\,P$
for a formula P. These two expressions (using the definitions above) are read as "there exists a friend of Peter who likes to dance" and "all friends of Peter like to dance", respectively. Variant notations include, for set X and set members x:
$\bigvee _{x}P$ $(\exists {x})P$[5] $(\exists x\ .\ P)$ $\exists x\ \cdot \ P$ $(\exists x:P)$ $\exists {x}(P)$[6] $\exists _{x}\,P$ $\exists {x}{,}\,P$ $\exists {x}{\in }X\,P$ $\exists \,x{:}X\,P$
All of these variations also apply to universal quantification. Other variations for the universal quantifier are
$\bigwedge _{x}P$ $\bigwedge xP$[7] $(x)\,P$[8]
Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; for a given mathematical theory, this can be done in several ways:
• Assume a fixed domain of discourse for every quantification, as is done in Zermelo–Fraenkel set theory,
• Fix several domains of discourse in advance and require that each variable have a declared domain, which is the type of that variable. This is analogous to the situation in statically typed computer programming languages, where variables have declared types.
• Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain (or the type of the objects in that domain).
One can use any variable as a quantified variable in place of any other, under certain restrictions in which variable capture does not occur. Even if the notation uses typed variables, variables of that type may be used.
Informally or in natural language, the "∀x" or "∃x" might appear after or in the middle of P(x). Formally, however, the phrase that introduces the dummy variable is placed in front.
Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as,
For every natural number x, ...
There exists an x such that ...
For at least one x, ....
Keywords for uniqueness quantification include:
For exactly one natural number x, ...
There is one and only one x such that ....
Further, x may be replaced by a pronoun. For example,
For every natural number, its product with 2 equals to its sum with itself.
Some natural number is prime.
Order of quantifiers (nesting)
The order of quantifiers is critical to meaning, as is illustrated by the following two propositions:
For every natural number n, there exists a natural number s such that s = n2.
This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the order of quantifiers is inversed is different:
There exists a natural number s such that for every natural number n, s = n2.
This is clearly false; it asserts that there is a single natural number s that is the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables.
A less trivial example from mathematical analysis regards the concepts of uniform and pointwise continuity, whose definitions differ only by an exchange in the positions of two quantifiers. A function f from R to R is called
• Pointwise continuous if
$\forall \varepsilon >0\;\forall x\in \mathbb {R} \;\exists \delta >0\;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )$
• Uniformly continuous if
$\forall \varepsilon >0\;\exists \delta >0\;\forall x\in \mathbb {R} \;\forall h\in \mathbb {R} \;(|h|<\delta \,\Rightarrow \,|f(x)-f(x+h)|<\varepsilon )$
In the former case, the particular value chosen for δ can be a function of both ε and x, the variables that precede it. In the latter case, δ can be a function only of ε (i.e., it has to be chosen independent of x). For example, f(x) = x2 satisfies pointwise, but not uniform continuity (its slope is unbound). In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning.
As a general rule, swapping two adjacent universal quantifiers with the same scope (or swapping two adjacent existential quantifiers with the same scope) doesn't change the meaning of the formula (see Example here), but swapping an existential quantifier and an adjacent universal quantifier may change its meaning.
The maximum depth of nesting of quantifiers in a formula is called its "quantifier rank".
Equivalent expressions
If D is a domain of x and P(x) is a predicate dependent on object variable x, then the universal proposition can be expressed as
$\forall x\!\in \!D\;P(x).$
This notation is known as restricted or relativized or bounded quantification. Equivalently one can write,
$\forall x\;(x\!\in \!D\to P(x)).$
The existential proposition can be expressed with bounded quantification as
$\exists x\!\in \!D\;P(x),$
or equivalently
$\exists x\;(x\!\in \!\!D\land P(x)).$
Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:
$\neg (\forall x\!\in \!D\;P(x))\equiv \exists x\!\in \!D\;\neg P(x),$
which shows that to disprove a "for all x" proposition, one needs no more than to find an x for which the predicate is false. Similarly,
$\neg (\exists x\!\in \!D\;P(x))\equiv \forall x\!\in \!D\;\neg P(x),$
to disprove a "there exists an x" proposition, one needs to show that the predicate is false for all x.
In classical logic, every formula is logically equivalent to a formula in prenex normal form, that is, a string of quantifiers and bound variables followed by a quantifier-free formula.
Quantifier elimination
This section is an excerpt from Quantifier elimination.[edit]
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "$\exists x$ such that $\ldots $" can be viewed as a question "When is there an $x$ such that $\ldots $?", and the statement without quantifiers can be viewed as the answer to that question.[9]
One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest.
A theory has quantifier elimination if for every formula $\alpha $, there exists another formula $\alpha _{QF}$ without quantifiers that is equivalent to it (modulo this theory).
Range of quantification
Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, say, asserting that a predicate holds for some natural number or for some real number. Expository conventions often reserve some variable names such as "n" for natural numbers, and "x" for real numbers, although relying exclusively on naming conventions cannot work in general, since ranges of variables can change in the course of a mathematical argument.
A universally quantified formula over an empty range (like $\forall x\!\in \!\varnothing \;x\neq x$) is always vacuously true. Conversely, an existentially quantified formula over an empty range (like $\exists x\!\in \!\varnothing \;x=x$) is always false.
A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification
For some natural number n, n is even and n is prime
means
For some even number n, n is prime.
In some mathematical theories, a single domain of discourse fixed in advance is assumed. For example, in Zermelo–Fraenkel set theory, variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above, to express
For every natural number n, n·2 = n + n
in Zermelo–Fraenkel set theory, one would write
For every n, if n belongs to N, then n·2 = n + n,
where N is the set of all natural numbers.
Formal semantics
Mathematical semantics is the application of mathematics to study the meaning of expressions in a formal language. It has three elements: a mathematical specification of a class of objects via syntax, a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. This article only addresses the issue of how quantifier elements are interpreted. The syntax of a formula can be given by a syntax tree. A quantifier has a scope, and an occurrence of a variable x is free if it is not within the scope of a quantification for that variable. Thus in
$\forall x(\exists yB(x,y))\vee C(y,x)$
the occurrence of both x and y in C(y, x) is free, while the occurrence of x and y in B(y, x) is bound (i.e. non-free).
An interpretation for first-order predicate calculus assumes as given a domain of individuals X. A formula A whose free variables are x1, ..., xn is interpreted as a boolean-valued function F(v1, ..., vn) of n arguments, where each argument ranges over the domain X. Boolean-valued means that the function assumes one of the values T (interpreted as truth) or F (interpreted as falsehood). The interpretation of the formula
$\forall x_{n}A(x_{1},\ldots ,x_{n})$
is the function G of n-1 arguments such that G(v1, ..., vn-1) = T if and only if F(v1, ..., vn-1, w) = T for every w in X. If F(v1, ..., vn-1, w) = F for at least one value of w, then G(v1, ..., vn-1) = F. Similarly the interpretation of the formula
$\exists x_{n}A(x_{1},\ldots ,x_{n})$
is the function H of n-1 arguments such that H(v1, ..., vn-1) = T if and only if F(v1, ..., vn-1, w) = T for at least one w and H(v1, ..., vn-1) = F otherwise.
The semantics for uniqueness quantification requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on X. The interpretation of
$\exists !x_{n}A(x_{1},\ldots ,x_{n})$
then is the function of n-1 arguments, which is the logical and of the interpretations of
$\exists x_{n}A(x_{1},\ldots ,x_{n})$
$\forall y,z{\big (}A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z{\big )}.$
Each kind of quantification defines a corresponding closure operator on the set of formulas, by adding, for each free variable x, a quantifier to bind x.[10] For example, the existential closure of the open formula n>2 ∧ xn+yn=zn is the closed formula ∃n ∃x ∃y ∃z (n>2 ∧ xn+yn=zn); the latter formula, when interpreted over the positive integers, is known to be false by Fermat's Last Theorem. As another example, equational axioms, like x+y=y+x, are usually meant to denote their universal closure, like ∀x ∀y (x+y=y+x) to express commutativity.
Paucal, multal and other degree quantifiers
See also: Fubini's theorem and measurable
None of the quantifiers previously discussed apply to a quantification such as
There are many integers n < 100, such that n is divisible by 2 or 3 or 5.
One possible interpretation mechanism can be obtained as follows: Suppose that in addition to a semantic domain X, we have given a probability measure P defined on X and cutoff numbers 0 < a ≤ b ≤ 1. If A is a formula with free variables x1,...,xn whose interpretation is the function F of variables v1,...,vn then the interpretation of
$\exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})$
is the function of v1,...,vn-1 which is T if and only if
$\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b$
and F otherwise. Similarly, the interpretation of
$\exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})$
is the function of v1,...,vn-1 which is F if and only if
$0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a$
and T otherwise.
Other quantifiers
A few other quantifiers have been proposed over time. In particular, the solution quantifier,[11]: 28 noted § (section sign) and read "those". For example,
$\left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}$
is read "those n in N such that n2 ≤ 4 are in {0,1,2}." The same construct is expressible in set-builder notation as
$\{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.$
Contrary to the other quantifiers, § yields a set rather than a formula.[12]
Some other quantifiers sometimes used in mathematics include:
• There are infinitely many elements such that...
• For all but finitely many elements... (sometimes expressed as "for almost all elements...").
• There are uncountably many elements such that...
• For all but countably many elements...
• For all elements in a set of positive measure...
• For all elements except those in a set of measure zero...
History
Term logic, also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated All, Some and No in the 4th century BC, in an account also touching on the alethic modalities.
In 1827, George Bentham published his Outline of a new system of logic, with a critical examination of Dr Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated.[13]
William Hamilton claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. 1840. Augustus De Morgan confirmed this in 1847, but modern usage began with De Morgan in 1862 where he makes statements such as "We are to take in both all and some-not-all as quantifiers".[14]
Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀x~, or contraposition. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics.
In work that culminated in Peirce (1885), Charles Sanders Peirce and his student Oscar Howard Mitchell independently invented universal and existential quantifiers, and bound variables. Peirce and Mitchell wrote Πx and Σx where we now write ∀x and ∃x. Peirce's notation can be found in the writings of Ernst Schröder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. Most notably, it is the notation of Kurt Gödel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic.
Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano, who invented yet another notation, namely (x) for the universal quantification of x and (in 1897) ∃x for the existential quantification of x. Hence for decades, the canonical notation in philosophy and mathematical logic was (x)P to express "all individuals in the domain of discourse have the property P," and "(∃x)P" for "there exists at least one individual in the domain of discourse having the property P." Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church. In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s.
Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching heterogeneous reasoning and diagrammatic inference.
See also
• Absolute generality
• Almost all
• Branching quantifier
• Conditional quantifier
• Counting quantification
• Eventually (mathematics)
• Generalized quantifier — a higher-order property used as standard semantics of quantified noun phrases
• Lindström quantifier — a generalized polyadic quantifier
• Quantifier shift
References
1. Kashef, Arman. (2023), In Quest of Univeral Logic: A brief overview of formal logic's evolution, doi:10.13140/RG.2.2.24043.82724/1
2. "Predicates and Quantifiers". www.csm.ornl.gov. Retrieved 2020-09-04.
3. "1.2 Quantifiers". www.whitman.edu. Retrieved 2020-09-04.
4. K.R. Apt (1990). "Logic Programming". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 493–574. ISBN 0-444-88074-7. Here: p.497
5. Schwichtenberg, Helmut; Wainer, Stanley S. (2009). Proofs and Computations. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139031905. ISBN 978-1-139-03190-5.
6. John E. Hopcroft and Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Reading/MA: Addison-Wesley. ISBN 0-201-02988-X. Here: p.p.344
7. Hans Hermes (1973). Introduction to Mathematical Logic. Hochschultext (Springer-Verlag). London: Springer. ISBN 3540058192. ISSN 1431-4657. Here: Def. II.1.5
8. Glebskii, Yu. V.; Kogan, D. I.; Liogon'kii, M. I.; Talanov, V. A. (1972). "Range and degree of realizability of formulas in the restricted predicate calculus". Cybernetics. 5 (2): 142–154. doi:10.1007/bf01071084. ISSN 0011-4235. S2CID 121409759.
9. Brown 2002.
10. in general, for a quantifer Q, closure makes sense only if the order of Q quantification does not matter, i.e. if Qx Qy p(x,y) is equivalent to Qy Qx p(x,y). This is satisfied for Q ∈ {∀,∃}, cf. #Order of quantifiers (nesting) above.
11. Hehner, Eric C. R., 2004, Practical Theory of Programming, 2nd edition, p. 28
12. Hehner (2004) uses the term "quantifier" in a very general sense, also including e.g. summation.
13. George Bentham, Outline of a new system of logic: with a critical examination of Dr. Whately's Elements of Logic (1827); Thoemmes; Facsimile edition (1990) ISBN 1-85506-029-9
14. Peters, Stanley; Westerståhl, Dag (2006-04-27). Quantifiers in Language and Logic. Clarendon Press. pp. 34–. ISBN 978-0-19-929125-0.
Bibliography
• Barwise, Jon; and Etchemendy, John, 2000. Language Proof and Logic. CSLI (University of Chicago Press) and New York: Seven Bridges Press. A gentle introduction to first-order logic by two first-rate logicians.
• Brown, Christopher W. (July 31, 2002). "What is Quantifier Elimination". Retrieved Aug 30, 2018.
• Frege, Gottlob, 1879. Begriffsschrift. Translated in Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book on Mathematical Logic, 1879-1931. Harvard University Press. The first appearance of quantification.
• Hilbert, David; and Ackermann, Wilhelm, 1950 (1928). Principles of Mathematical Logic. Chelsea. Translation of Grundzüge der theoretischen Logik. Springer-Verlag. The 1928 first edition is the first time quantification was consciously employed in the now-standard manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of first-order logic.
• Peirce, C. S., 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, American Journal of Mathematics, Vol. 7, pp. 180–202. Reprinted in Kloesel, N. et al., eds., 1993. Writings of C. S. Peirce, Vol. 5. Indiana University Press. The first appearance of quantification in anything like its present form.
• Reichenbach, Hans, 1975 (1947). Elements of Symbolic Logic, Dover Publications. The quantifiers are discussed in chapters §18 "Binding of variables" through §30 "Derivations from Synthetic Premises".
• Westerståhl, Dag, 2001, "Quantifiers," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• Wiese, Heike, 2003. Numbers, language, and the human mind. Cambridge University Press. ISBN 0-521-83182-2.
External links
Look up quantification in Wiktionary, the free dictionary.
• "Quantifier", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• ""For all" and "there exists" topical phrases, sentences and expressions". Archived from the original on March 1, 2000.. From College of Natural Sciences, University of Hawaii at Manoa.
• Stanford Encyclopedia of Philosophy:
• Shapiro, Stewart (2000). "Classical Logic" (Covers syntax, model theory, and metatheory for first order logic in the natural deduction style.)
• Westerståhl, Dag (2005). "Generalized quantifiers"
• Peters, Stanley; Westerståhl, Dag (2002). "Quantifiers"
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Wikipedia
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Q–Q plot
In statistics, a Q–Q plot (quantile-quantile plot) is a probability plot, a graphical method for comparing two probability distributions by plotting their quantiles against each other.[1] A point (x, y) on the plot corresponds to one of the quantiles of the second distribution (y-coordinate) plotted against the same quantile of the first distribution (x-coordinate). This defines a parametric curve where the parameter is the index of the quantile interval.
If the two distributions being compared are similar, the points in the Q–Q plot will approximately lie on the identity line y = x. If the distributions are linearly related, the points in the Q–Q plot will approximately lie on a line, but not necessarily on the line y = x. Q–Q plots can also be used as a graphical means of estimating parameters in a location-scale family of distributions.
A Q–Q plot is used to compare the shapes of distributions, providing a graphical view of how properties such as location, scale, and skewness are similar or different in the two distributions. Q–Q plots can be used to compare collections of data, or theoretical distributions. The use of Q–Q plots to compare two samples of data can be viewed as a non-parametric approach to comparing their underlying distributions. A Q–Q plot is generally more diagnostic than comparing the samples' histograms, but is less widely known. Q–Q plots are commonly used to compare a data set to a theoretical model.[2][3] This can provide an assessment of goodness of fit that is graphical, rather than reducing to a numerical summary statistic. Q–Q plots are also used to compare two theoretical distributions to each other.[4] Since Q–Q plots compare distributions, there is no need for the values to be observed as pairs, as in a scatter plot, or even for the numbers of values in the two groups being compared to be equal.
The term "probability plot" sometimes refers specifically to a Q–Q plot, sometimes to a more general class of plots, and sometimes to the less commonly used P–P plot. The probability plot correlation coefficient plot (PPCC plot) is a quantity derived from the idea of Q–Q plots, which measures the agreement of a fitted distribution with observed data and which is sometimes used as a means of fitting a distribution to data.
Definition and construction
A Q–Q plot is a plot of the quantiles of two distributions against each other, or a plot based on estimates of the quantiles. The pattern of points in the plot is used to compare the two distributions.
The main step in constructing a Q–Q plot is calculating or estimating the quantiles to be plotted. If one or both of the axes in a Q–Q plot is based on a theoretical distribution with a continuous cumulative distribution function (CDF), all quantiles are uniquely defined and can be obtained by inverting the CDF. If a theoretical probability distribution with a discontinuous CDF is one of the two distributions being compared, some of the quantiles may not be defined, so an interpolated quantile may be plotted. If the Q–Q plot is based on data, there are multiple quantile estimators in use. Rules for forming Q–Q plots when quantiles must be estimated or interpolated are called plotting positions.
A simple case is where one has two data sets of the same size. In that case, to make the Q–Q plot, one orders each set in increasing order, then pairs off and plots the corresponding values. A more complicated construction is the case where two data sets of different sizes are being compared. To construct the Q–Q plot in this case, it is necessary to use an interpolated quantile estimate so that quantiles corresponding to the same underlying probability can be constructed.
More abstractly,[4] given two cumulative probability distribution functions F and G, with associated quantile functions F−1 and G−1 (the inverse function of the CDF is the quantile function), the Q–Q plot draws the q-th quantile of F against the q-th quantile of G for a range of values of q. Thus, the Q–Q plot is a parametric curve indexed over [0,1] with values in the real plane R2.
Interpretation
The points plotted in a Q–Q plot are always non-decreasing when viewed from left to right. If the two distributions being compared are identical, the Q–Q plot follows the 45° line y = x. If the two distributions agree after linearly transforming the values in one of the distributions, then the Q–Q plot follows some line, but not necessarily the line y = x. If the general trend of the Q–Q plot is flatter than the line y = x, the distribution plotted on the horizontal axis is more dispersed than the distribution plotted on the vertical axis. Conversely, if the general trend of the Q–Q plot is steeper than the line y = x, the distribution plotted on the vertical axis is more dispersed than the distribution plotted on the horizontal axis. Q–Q plots are often arced, or "S" shaped, indicating that one of the distributions is more skewed than the other, or that one of the distributions has heavier tails than the other.
Although a Q–Q plot is based on quantiles, in a standard Q–Q plot it is not possible to determine which point in the Q–Q plot determines a given quantile. For example, it is not possible to determine the median of either of the two distributions being compared by inspecting the Q–Q plot. Some Q–Q plots indicate the deciles to make determinations such as this possible.
The intercept and slope of a linear regression between the quantiles gives a measure of the relative location and relative scale of the samples. If the median of the distribution plotted on the horizontal axis is 0, the intercept of a regression line is a measure of location, and the slope is a measure of scale. The distance between medians is another measure of relative location reflected in a Q–Q plot. The "probability plot correlation coefficient" (PPCC plot) is the correlation coefficient between the paired sample quantiles. The closer the correlation coefficient is to one, the closer the distributions are to being shifted, scaled versions of each other. For distributions with a single shape parameter, the probability plot correlation coefficient plot provides a method for estimating the shape parameter – one simply computes the correlation coefficient for different values of the shape parameter, and uses the one with the best fit, just as if one were comparing distributions of different types.
Another common use of Q–Q plots is to compare the distribution of a sample to a theoretical distribution, such as the standard normal distribution N(0,1), as in a normal probability plot. As in the case when comparing two samples of data, one orders the data (formally, computes the order statistics), then plots them against certain quantiles of the theoretical distribution.[3]
Plotting positions
The choice of quantiles from a theoretical distribution can depend upon context and purpose. One choice, given a sample of size n, is k / n for k = 1, …, n, as these are the quantiles that the sampling distribution realizes. The last of these, n / n, corresponds to the 100th percentile – the maximum value of the theoretical distribution, which is sometimes infinite. Other choices are the use of (k − 0.5) / n, or instead to space the points evenly in the uniform distribution, using k / (n + 1).[6]
Many other choices have been suggested, both formal and heuristic, based on theory or simulations relevant in context. The following subsections discuss some of these. A narrower question is choosing a maximum (estimation of a population maximum), known as the German tank problem, for which similar "sample maximum, plus a gap" solutions exist, most simply m + m/n − 1. A more formal application of this uniformization of spacing occurs in maximum spacing estimation of parameters.
Expected value of the order statistic for a uniform distribution
The k / (n + 1) approach equals that of plotting the points according to the probability that the last of (n + 1) randomly drawn values will not exceed the k-th smallest of the first n randomly drawn values.[7][8]
Expected value of the order statistic for a standard normal distribution
In using a normal probability plot, the quantiles one uses are the rankits, the quantile of the expected value of the order statistic of a standard normal distribution.
More generally, Shapiro–Wilk test uses the expected values of the order statistics of the given distribution; the resulting plot and line yields the generalized least squares estimate for location and scale (from the intercept and slope of the fitted line).[9] Although this is not too important for the normal distribution (the location and scale are estimated by the mean and standard deviation, respectively), it can be useful for many other distributions.
However, this requires calculating the expected values of the order statistic, which may be difficult if the distribution is not normal.
Median of the order statistics
Alternatively, one may use estimates of the median of the order statistics, which one can compute based on estimates of the median of the order statistics of a uniform distribution and the quantile function of the distribution; this was suggested by (Filliben 1975).[9]
This can be easily generated for any distribution for which the quantile function can be computed, but conversely the resulting estimates of location and scale are no longer precisely the least squares estimates, though these only differ significantly for n small.
Heuristics
Several different formulas have been used or proposed as affine symmetrical plotting positions. Such formulas have the form (k − a) / (n + 1 − 2a) for some value of a in the range from 0 to 1, which gives a range between k / (n + 1) and (k − 1) / (n − 1).
Expressions include:
• k / (n + 1)
• (k − 0.3) / (n + 0.4).[10]
• (k − 0.3175) / (n + 0.365).[11][note 1]
• (k − 0.326) / (n + 0.348).[12]
• (k − ⅓) / (n + ⅓).[note 2]
• (k − 0.375) / (n + 0.25).[note 3]
• (k − 0.4) / (n + 0.2).[13]
• (k − 0.44) / (n + 0.12).[note 4]
• (k − 0.5) / n.[15]
• (k − 0.567) / (n − 0.134).[16]
• (k − 1) / (n − 1).[note 5]
For large sample size, n, there is little difference between these various expressions.
Filliben's estimate
The order statistic medians are the medians of the order statistics of the distribution. These can be expressed in terms of the quantile function and the order statistic medians for the continuous uniform distribution by:
$N(i)=G(U(i))$
where U(i) are the uniform order statistic medians and G is the quantile function for the desired distribution. The quantile function is the inverse of the cumulative distribution function (probability that X is less than or equal to some value). That is, given a probability, we want the corresponding quantile of the cumulative distribution function.
James J. Filliben (Filliben 1975) uses the following estimates for the uniform order statistic medians:
$m(i)={\begin{cases}1-0.5^{1/n}&i=1\\\\{\dfrac {i-0.3175}{n+0.365}}&i=2,3,\ldots ,n-1\\\\0.5^{1/n}&i=n.\end{cases}}$
The reason for this estimate is that the order statistic medians do not have a simple form.
See also
• Empirical distribution function
• Probit analysis was developed by Chester Ittner Bliss in 1934.
Notes
1. Note that this also uses a different expression for the first & last points. cites the original work by (Filliben 1975). This expression is an estimate of the medians of U(k).
2. A simple (and easy to remember) formula for plotting positions; used in BMDP statistical package.
3. This is (Blom 1958)'s earlier approximation and is the expression used in MINITAB.
4. This plotting position was used by Irving I. Gringorten[14] to plot points in tests for the Gumbel distribution.
5. Used by Filliben (1975), these plotting points are equal to the modes of U(k).
References
Citations
1. Wilk, M.B.; Gnanadesikan, R. (1968), "Probability plotting methods for the analysis of data", Biometrika, Biometrika Trust, 55 (1): 1–17, doi:10.1093/biomet/55.1.1, JSTOR 2334448, PMID 5661047.
2. Gnanadesikan (1977) p199.
3. (Thode 2002, Section 2.2.2, Quantile-Quantile Plots, p. 21)
4. (Gibbons & Chakraborti 2003, p. 144)
5. "SR 20 – North Cascades Highway – Opening and Closing History". North Cascades Passes. Washington State Department of Transportation. October 2009. Retrieved 8 February 2009.
6. Weibull, Waloddi (1939), "The Statistical Theory of the Strength of Materials", IVA Handlingar, Royal Swedish Academy of Engineering Sciences (151)
7. Madsen, H.O.; et al. (1986), Methods of Structural Safety
8. Makkonen, L. (2008), "Bringing closure to the plotting position controversy", Communications in Statistics – Theory and Methods, 37 (3): 460–467, doi:10.1080/03610920701653094, S2CID 122822135
9. Testing for Normality, by Henry C. Thode, CRC Press, 2002, ISBN 978-0-8247-9613-6, p. 31
10. Benard & Bos-Levenbach (1953) harvtxt error: no target: CITEREFBenardBos-Levenbach1953 (help). The plotting of observations on probability paper. Statistica Neederlandica, 7: 163-173. doi:10.1111/j.1467-9574.1953.tb00821.x. (in Dutch)
11. "1.3.3.21. Normal Probability Plot". itl.nist.gov. Retrieved 16 February 2022.
12. Distribution free plotting position, Yu & Huang
13. Cunnane (1978) harvtxt error: no target: CITEREFCunnane1978 (help).
14. Gringorten, Irving I. (1963). "A plotting rule for extreme probability paper". Journal of Geophysical Research. 68 (3): 813–814. Bibcode:1963JGR....68..813G. doi:10.1029/JZ068i003p00813. ISSN 2156-2202.
15. Hazen, Allen (1914), "Storage to be provided in the impounding reservoirs for municipal water supply", Transactions of the American Society of Civil Engineers (77): 1547–1550
16. Larsen, Curran & Hunt (1980) harvtxt error: no target: CITEREFLarsenCurranHunt1980 (help).
Sources
• This article incorporates public domain material from the National Institute of Standards and Technology.
• Blom, G. (1958), Statistical estimates and transformed beta variables, New York: John Wiley and Sons
• Chambers, John; William Cleveland; Beat Kleiner; Paul Tukey (1983), Graphical methods for data analysis, Wadsworth
• Cleveland, W.S. (1994) The Elements of Graphing Data, Hobart Press ISBN 0-9634884-1-4
• Filliben, J. J. (February 1975), "The Probability Plot Correlation Coefficient Test for Normality", Technometrics, American Society for Quality, 17 (1): 111–117, doi:10.2307/1268008, JSTOR 1268008.
• Gibbons, Jean Dickinson; Chakraborti, Subhabrata (2003), Nonparametric statistical inference (4th ed.), CRC Press, ISBN 978-0-8247-4052-8
• Gnanadesikan, R. (1977) Methods for Statistical Analysis of Multivariate Observations, Wiley ISBN 0-471-30845-5.
• Thode, Henry C. (2002), Testing for normality, New York: Marcel Dekker, ISBN 0-8247-9613-6
External links
Wikimedia Commons has media related to Q-Q plot.
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Quantile regression averaging
Quantile Regression Averaging (QRA) is a forecast combination approach to the computation of prediction intervals. It involves applying quantile regression to the point forecasts of a small number of individual forecasting models or experts. It has been introduced in 2014 by Jakub Nowotarski and Rafał Weron[1] and originally used for probabilistic forecasting of electricity prices[2][3] and loads.[4][5] Despite its simplicity it has been found to perform extremely well in practice - the top two performing teams in the price track of the Global Energy Forecasting Competition (GEFCom2014) used variants of QRA.[6][7]
Introduction
The individual point forecasts are used as independent variables and the corresponding observed target variable as the dependent variable in a standard quantile regression setting.[8] The Quantile Regression Averaging method yields an interval forecast of the target variable, but does not use the prediction intervals of the individual methods. One of the reasons for using point forecasts (and not interval forecasts) is their availability. For years, forecasters have focused on obtaining accurate point predictions. Computing probabilistic forecasts, on the other hand, is generally a much more complex task and has not been discussed in the literature nor developed by practitioners so extensively. Therefore, QRA may be found particularly attractive from a practical point of view as it allows to leverage existing development of point forecasting.
Computation
The quantile regression problem can be written as follows:
$Q_{y}(q|X_{t})=X_{t}\beta _{q}$,
where $Q_{y}(q|\cdot )$ is the conditional q-th quantile of the dependent variable ($y_{t}$), $X_{t}=[1,{\hat {y}}_{1,t},...,{\hat {y}}_{m,t}]$ is a vector of point forecasts of $m$ individual models (i.e. independent variables) and βq is a vector of parameters (for quantile q). The parameters are estimated by minimizing the loss function for a particular q-th quantile:
$\min \limits _{\beta _{q}}\left[\sum \limits _{\{t:y_{t}\geq X_{t}\beta _{q}\}}q|y_{t}-X_{t}\beta _{q}|+\sum \limits _{\{t:y_{t}<X_{t}\beta _{q}\}}(1-q)|y_{t}-X_{t}\beta _{q}|\right]=\min \limits _{\beta _{q}}\left[\sum \limits _{t}(q-\mathbf {1} _{y_{t}<X_{t}\beta _{q}})(y_{t}-X_{t}\beta _{q})\right]$
QRA assigns weights to individual forecasting methods and combines them to yield forecasts of chosen quantiles. Although the QRA method is based on quantile regression, not least squares, it still suffers from the same problems: the exogenous variables should not be correlated strongly and the number of variables included in the model has to be relatively small in order for the method to be computationally efficient.
Factor Quantile Regression Averaging (FQRA)
The main difficulty associated with applying QRA comes from the fact that only individual models that perform well and (preferably) are distinct should be used. However, there may be many well performing models or many different specifications of each model (with or without exogenous variables, with all or only selected lags, etc.) and it may not be optimal to include all of them in Quantile Regression Averaging.
In Factor Quantile Regression Averaging (FQRA),[3] instead of selecting individual models a priori, the relevant information contained in all forecasting models at hand is extracted using principal component analysis (PCA). The prediction intervals are then constructed on the basis of the common factors ($f_{t}$) obtained from the panel of point forecasts, as independent variables in a quantile regression. More precisely, in the FQRA method $X_{t}=[1,{\hat {f}}_{1,t},...,{\hat {f}}_{k,t}]$ is a vector of $k<m$ factors extracted from a panel of point forecasts of $m$ individual models, not a vector of point forecasts of the individual models themselves. A similar principal component-type approach was proposed in the context of obtaining point forecasts from the Survey of Professional Forecasters data.[9]
Instead of considering a (large) panel of forecasts of the individual models, FQRA concentrates on a small number of common factors, which - by construction - are orthogonal to each other, and hence are contemporaneously uncorrelated. FQRA can be also interpreted as a forecast averaging approach. The factors estimated within PCA are linear combinations of individual vectors of the panel and FQRA can therefore be used to assign weights to the forecasting models directly.
QRA and LAD regression
QRA may be viewed as an extension of combining point forecasts. The well-known ordinary least squares (OLS) averaging[10] uses linear regression to estimate weights of the point forecasts of individual models. Replacing the quadratic loss function with the absolute loss function leads to quantile regression for the median, or in other words, least absolute deviation (LAD) regression.[11]
See also
• Consensus forecast, also known as combining forecasts, forecast averaging or model averaging (in econometrics and statistics) and committee machines, ensemble averaging or expert aggregation (in machine learning)
• Electricity price forecasting
• Energy forecasting
• Forecasting
• Global Energy Forecasting Competitions
• Economic forecasting
• Prediction interval
• Probabilistic forecasting
• Quantile regression
Implementations
• Matlab code for computing interval forecasts using QRA is available from RePEc: https://ideas.repec.org/c/wuu/hscode/m14003.html
References
1. Nowotarski, Jakub; Weron, Rafał (2015). [Open Access]. "Computing electricity spot price prediction intervals using quantile regression and forecast averaging". Computational Statistics. 30 (3): 791–803. doi:10.1007/s00180-014-0523-0. ISSN 0943-4062.
2. Weron, Rafał (2014). [Open Access]. "Electricity price forecasting: A review of the state-of-the-art with a look into the future". International Journal of Forecasting. 30 (4): 1030–1081. doi:10.1016/j.ijforecast.2014.08.008.
3. Maciejowska, Katarzyna; Nowotarski, Jakub; Weron, Rafał (2016). "Probabilistic forecasting of electricity spot prices using Factor Quantile Regression Averaging". International Journal of Forecasting. 32 (3): 957–965. doi:10.1016/j.ijforecast.2014.12.004.
4. Liu, B.; Nowotarski, J.; Hong, T.; Weron, R. (2015). "Probabilistic Load Forecasting via Quantile Regression Averaging on Sister Forecasts". IEEE Transactions on Smart Grid. PP (99): 1. doi:10.1109/TSG.2015.2437877. ISSN 1949-3053.
5. Hong, Tao; Fan, Shu. "Probabilistic Electric Load Forecasting: A Tutorial Review". blog.drhongtao.com. Retrieved 2015-11-28.
6. Gaillard, Pierre; Goude, Yannig; Nedellec, Raphaël (2016). "Additive models and robust aggregation for GEFCom2014 probabilistic electric load and electricity price forecasting". International Journal of Forecasting. 32 (3): 1038–1050. doi:10.1016/j.ijforecast.2015.12.001.
7. Maciejowska, Katarzyna; Nowotarski, Jakub (2016). "A hybrid model for GEFCom2014 probabilistic electricity price forecasting" (PDF). International Journal of Forecasting. 32 (3): 1051–1056. doi:10.1016/j.ijforecast.2015.11.008.
8. Koenker, Roger (2005). "Quantile Regression This article has been prepared for the Statistical Theory and Methods section of the Encyclopedia of Environmetrics edited by Abdel El-Shaarawi and Walter Piegorsch. The research was partially supported by NSF grant SES-0850060". Quantile Regression. John Wiley & Sons, Ltd. doi:10.1002/9780470057339.vnn091. ISBN 9780470057339.
9. Poncela, Pilar; Rodríguez, Julio; Sánchez-Mangas, Rocío; Senra, Eva (2011). "Forecast combination through dimension reduction techniques". International Journal of Forecasting. 27 (2): 224–237. doi:10.1016/j.ijforecast.2010.01.012.
10. Granger, Clive W. J.; Ramanathan, Ramu (1984). "Improved methods of combining forecasts". Journal of Forecasting. 3 (2): 197–204. doi:10.1002/for.3980030207. ISSN 1099-131X.
11. Nowotarski, Jakub; Raviv, Eran; Trück, Stefan; Weron, Rafał (2014). "An empirical comparison of alternative schemes for combining electricity spot price forecasts". Energy Economics. 46: 395–412. doi:10.1016/j.eneco.2014.07.014.
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Wikipedia
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Quantitative psychology
Quantitative psychology is a field of scientific study that focuses on the mathematical modeling, research design and methodology, and statistical analysis of psychological processes. It includes tests and other devices for measuring cognitive abilities. Quantitative psychologists develop and analyze a wide variety of research methods, including those of psychometrics, a field concerned with the theory and technique of psychological measurement.[1]
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Psychologists have long contributed to statistical and mathematical analysis, and quantitative psychology is now a specialty recognized by the American Psychological Association. Doctoral degrees are awarded in this field in a number of universities in Europe and North America, and quantitative psychologists have been in high demand in industry, government, and academia. Their training in both social science and quantitative methodology provides a unique skill set for solving both applied and theoretical problems in a variety of areas.
History
Quantitative psychology has its roots in early experimental psychology when, in the nineteenth century, the scientific method was first systematically applied to psychological phenomena. Notable contributions included E. H. Weber's studies of tactile sensitivity (1830s), Fechner's development and use of the psychophysical methods (1850-1860), and Helmholtz's research on vision and audition beginning after 1850. Wilhelm Wundt is often called the "founder of experimental psychology", because he called himself a psychologist and opened a psychological laboratory in 1879 where many researchers came to study.[2] The work of these and many others helped put to rest the assertion, by theorists such as Immanuel Kant, that psychology could not become a science because precise experiments on the human mind were impossible.
Intelligence testing
Intelligence testing has long been an important branch of quantitative psychology. The nineteenth-century English statistician Francis Galton, a pioneer in psychometrics, was the first to create a standardized test of intelligence, and he was among the first to apply statistical methods to the study of human differences and their inheritance. He came to believe that intelligence is largely determined by heredity, and he also hypothesized that other measures such as the speed of reflexes, muscle strength, and head size are correlated with intelligence.[3][4] He established the world's first mental testing center in 1882; in the following year he published his observations and theories in "Inquiries into Human Faculty and Its Development".
Statistical techniques
Statistical methods are the quantitative tools most used by psychologists. Pearson introduced the correlation coefficient and the chi-squared test. The 1900–1920 period saw the t-test (Student, 1908), the ANOVA (Fisher, 1925) and a non-parametric correlation coefficient (Spearman, 1904). A large number of tests were developed in the latter half of the 20th century (e.g., all the multivariate tests). Popular techniques (such as Hierarchical Linear Model, Arnold, 1992, Structural Equation Modeling, Byrne, 1996 and Independent Component Analysis, Hyvarinën, Karhunen and Oja, 2001) are relatively recent.[5]
In 1946, psychologist Stanley Smith Stevens organized levels of measurement into four scales, Nominal, Ordinal, Ratio, and Interval, in a paper that is still often cited.[6] Jacob Cohen, a New York University professor of psychology, analyzed quantitative methods involving statistical power and effect size, which helped to lay foundations for current statistical meta-analysis and the methods of estimation statistics.[7] He gave his name to Cohen's kappa and Cohen's d.
In 1990, an influential paper titled "Graduate Training in Statistics, Methodology, and Measurement in Psychology" was published in the American Psychologist journal. This article discussed the need for increased and up-to-date training in quantitative methods for psychology graduate programs in the United States.[8]
Education and training
Undergraduate
Training for quantitative psychology can begin informally at the undergraduate level. Many graduate schools recommend that students have some coursework in psychology and complete the full college sequence of calculus (including multivariate calculus) and a course in linear algebra. Quantitative coursework in other fields such as economics and research methods and statistics courses for psychology majors are also helpful. Historically, however, students without all these courses have been accepted if other aspects of their application show promise. Some schools also offer formal minors in areas related to quantitative psychology. For example, the University of Kansas offers a minor in "Social and Behavioral Sciences Methodology" that provides advanced training in research methodology, applied data analysis, and practical research experience relevant to quantitative psychology.[9] Coursework in computer science is also useful. Mastery of an object-oriented programming language or learning to write code in R, SAS, or SPSS is useful for the type of data analysis performed in graduate school.
Graduate
Quantitative psychologists may possess a doctoral degree or a master's degree. Due to its interdisciplinary nature and depending on the research focus of the university, these programs may be housed in a school's college of education or in their psychology department. Programs that focus especially in educational research and psychometrics are often part of education or educational psychology departments. These programs may therefore have different names mentioning "research methods" or "quantitative methods", such as the "Research and Evaluation Methodology" Ph.D. from the University of Florida or the "Quantitative Methods" degree at the University of Pennsylvania. However, some universities may have separate programs in their two colleges. For example, the University of Washington has a "Quantitative psychology" degree in their psychology department and a separate "Measurement & Statistics" Ph.D. in their college of education. Others, such as Vanderbilt University's Ph.D. in Psychological Sciences, are jointly housed across two psychology departments.[10]
Universities with a mathematical focus include McGill University's "Quantitative Psychology and Modeling" program and Purdue University's "Mathematical and Computational Psychology" degrees.[11][12] Students with an interest in modeling biological or functional data may go into related fields such as biostatistics or computational neuroscience.
Doctoral programs typically accept students with only bachelor's degrees, although some schools may require a master's degree before applying. After the first two years of studies, graduate students typically earn a Master of Arts in Psychology, Master of Science in Statistics or Applied statistics, or both. For example, most students in the University of Minnesota's "Quantitative and Psychometric Methods" Ph.D. program are also Master of Science students in the School of Statistics.[13] Additionally, several universities offer minor concentrations in quantitative methods, such as New York University.[14]
Companies that produce standardized tests such as College Board, Educational Testing Service, and American College Testing are some of the largest private sector employers of quantitative psychologists. These companies also often provide internships to students in graduate school.
Shortage of qualified applicants
In August 2005, the American Psychological Association expressed the need for more quantitative psychologists in the industry—for every PhD awarded in the subject, there were about 2.5 quantitative psychologist position openings.[15] Due to a lack of applicants in the field, the APA created a Task Force to study the state of quantitative psychology and predict its future. Domestic U.S. applicants are especially lacking. The majority of international applicants come from Asian countries, especially South Korea and China.[16] In response to the lack of qualified applicants, the APA Council of Representatives authorized a special task force in 2006.[17] The task force was chaired by Leona S. Aiken from Arizona State University.
Research areas
Quantitative psychologists generally have a main area of interest.[18] Notable research areas in psychometrics include item response theory and computer adaptive testing, which focus on education and intelligence testing. Other research areas include modeling psychological processes through time series analysis, such as in fMRI data collection, and structural equation modeling, social network analysis, human decision science, and statistical genetics.
Two common types of psychometric tests are aptitude tests, which are supposed to measure raw intellectual suitability for a purpose, and personality tests, which aim to assess character, temperament, and how the subject deals with problems.
Item response theory is based on the application of related mathematical models to testing data. Because it is generally regarded as superior to classical test theory, it is the preferred method for developing scales in the United States, especially when optimal decisions are demanded, as in so-called high-stakes tests, e.g., the Graduate Record Examination (GRE) and Graduate Management Admission Test (GMAT).
Professional organizations
Quantitative psychology is served by several scientific organizations. These include the Psychometric Society, Division 5 of the American Psychological Association (Evaluation, Measurement and Statistics), the Society of Multivariate Experimental Psychology, and the European Society for Methodology. Associated disciplines include statistics, mathematics, educational measurement, educational statistics, sociology, and political science. Several scholarly journals reflect the efforts of scientists in these areas, notably Psychometrika, Psychological Methods, Multivariate Behavioral Research, and Structural Equation Modeling.
Notable people
The following is a select list of quantitative psychologists or people who have contributed to the field:
• Leona S. Aiken
• Anne Anastasi
• Gwyneth Boodoo
• Raymond Cattell
• Jacob Cohen
• Lee Cronbach
• Louis Guttman
• Frederic M. Lord
• Quinn McNemar
• Paul E. Meehl
• Jacqueline Meulman
• Peter Molenaar
• Pip Pattison
• James O. Ramsay
• Robert L. Thorndike
• Louis Leon Thurstone
• Helen M. Walker
See also
• List of schools for quantitative psychology
• Mathematical psychology
• Measuring the Mind
• Network neuroscience
• Psychophysics
• Psychometrics
• Psychometrika
• Quantitative psychological research
• WinBUGS
References
1. "Classification of Instructional Programs – Psychometrics and Quantitative Psychology". The Integrated Postsecondary Education Data System. Retrieved 19 January 2015.
2. E. Hearst (ed) The First Century of Experimental Psychology, 1979, pp. 19-20, Hillsdale, NJ: Earlbaum
3. Bulmer, M. (1999). The development of Francis Galton's ideas on the mechanism of heredity. Journal of the History of Biology, 32(3), 263-292. Cowan, R. S. (1972). Francis Galton's contribution to genetics. Journal of the History of Biology, 5(2), 389-412. See also Burbridge, D. (2001). Francis Galton on twins, heredity and social class. British Journal for the History of Science, 34(3), 323-340.
4. Fancher, R. E. (1983). Biographical origins of Francis Galton's psychology. Isis, 74(2), 227-233.
5. Cousineau, Denis (2005). "The rise of quantitative methods in psychology". Tutorial in Quantitative Methods for Psychology. 1 (1): 1–3. doi:10.20982/tqmp.01.1.p001.
6. Stevens, Stanley Smith (June 7, 1946). "On the Theory of Scales of Measurement" (PDF). Science. 103 (2684): 677–680. Bibcode:1946Sci...103..677S. doi:10.1126/science.103.2684.677. PMID 17750512. Archived from the original (PDF) on September 6, 2012. Retrieved September 16, 2010.
7. Cohen's entry in Encyclopedia of Statistics in Behavioral Science
8. Aiken, Leona S.; West, Stephen G. (June 1990). "Graduate Training in Statistics, Methodology, and Measurement in Psychology: A Survey of PhD Programs in North America" (PDF). American Psychologist. 45 (6): 721–734. doi:10.1037/0003-066x.45.6.721. Archived from the original (PDF) on 2015-01-19. Retrieved 19 January 2015.
9. "Undergraduate Minor in Social and Behavioral Sciences Methodology". University of Kansas. Retrieved 13 December 2014.
10. "Quantitative Methods at Vanderbilt University". Vanderbilt University, Psychological Sciences. Retrieved 2022-05-05.
11. "Quantitative Psychology & Modelling". McGill University, Department of Psychology. Retrieved 2022-05-05.
12. "Mathematical and Computational Psychology". Purdue University, Psychological Sciences. Retrieved 2022-05-05.
13. "Quantitative/Psychometric Methods at University of Minnesota". University of Minnesota, College of Liberal Arts. Retrieved 2022-05-05.
14. "Quantitative Minor at New York University". New York University, Arts & Science. Retrieved 2022-05-05.
15. Report of the Task Force for Increasing the Number of Quantitative Psychologists, page 1. American Psychological Association. Retrieved February 15, 2012
16. "Report of the Task Force for Increasing the Number of Quantitative Psychologists" (PDF). American Psychological Association. Retrieved 13 December 2014.
17. "Quantitative Psychology". American Psychological Association. Retrieved 19 January 2015.
18. Mitchell J. Prinstein (31 August 2012). The Portable Mentor: Expert Guide to a Successful Career in Psychology. Springer Science & Business Media. p. 24. ISBN 978-1-4614-3993-6.
Further reading
• "Report of the Task Force for Increasing the Number of Quantitative Psychologists" (PDF). American Psychological Association. Retrieved 13 December 2014.
External links
• APA Division 5: Evaluation, Measurement and Statistics
• The Psychometric Society
• The Society of Multivariate Experimental Psychology
• The European Society for Methodology
• Society for Mathematical Psychology
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Quantization commutes with reduction
In mathematics, more specifically in the context of geometric quantization, quantization commutes with reduction states that the space of global sections of a line bundle L satisfying the quantization condition[1] on the symplectic quotient of a compact symplectic manifold is the space of invariant sections of L.
This was conjectured in 1980s by Guillemin and Sternberg and was proven in 1990s by Meinrenken[2][3] (the second paper used symplectic cut) as well as Tian and Zhang.[4] For the formulation due to Teleman, see C. Woodward's notes.
See also
• Geometric invariant theory
Notes
1. This means that the curvature of the connection on the line bundle is the symplectic form.
2. Meinrenken 1996
3. Meinrenken 1998
4. Tian & Zhang 1998
References
• Guillemin, V.; Sternberg, S. (1982), "Geometric quantization and multiplicities of group representations", Inventiones Mathematicae, 67 (3): 515–538, Bibcode:1982InMat..67..515G, doi:10.1007/BF01398934, MR 0664118, S2CID 121632102
• Meinrenken, Eckhard (1996), "On Riemann-Roch formulas for multiplicities", Journal of the American Mathematical Society, 9 (2): 373–389, doi:10.1090/S0894-0347-96-00197-X, MR 1325798.
• Meinrenken, Eckhard (1998), "Symplectic surgery and the Spinc-Dirac operator", Advances in Mathematics, 134 (2): 240–277, arXiv:dg-ga/9504002, doi:10.1006/aima.1997.1701, MR 1617809.
• Tian, Youliang; Zhang, Weiping (1998), "An analytic proof of the geometric quantization conjecture of Guillemin–Sternberg", Inventiones Mathematicae, 132 (2): 229–259, Bibcode:1998InMat.132..229T, doi:10.1007/s002220050223, MR 1621428, S2CID 119943992.
• Woodward, Christopher T. (2010), "Moment maps and geometric invariant theory", Les cours du CIRM, 1 (1): 55–98, arXiv:0912.1132, Bibcode:2009arXiv0912.1132W, doi:10.5802/ccirm.4
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Quantized state systems method
The quantized state systems (QSS) methods are a family of numerical integration solvers based on the idea of state quantization, dual to the traditional idea of time discretization. Unlike traditional numerical solution methods, which approach the problem by discretizing time and solving for the next (real-valued) state at each successive time step, QSS methods keep time as a continuous entity and instead quantize the system's state, instead solving for the time at which the state deviates from its quantized value by a quantum.
They can also have many advantages compared to classical algorithms.[1] They inherently allow for modeling discontinuities in the system due to their discrete-event nature and asynchronous nature. They also allow for explicit root-finding and detection of zero-crossing using explicit algorithms, avoiding the need for iteration---a fact which is especially important in the case of stiff systems, where traditional time-stepping methods require a heavy computational penalty due to the requirement to implicitly solve for the next system state. Finally, QSS methods satisfy remarkable global stability and error bounds, described below, which are not satisfied by classical solution techniques.
By their nature, QSS methods are therefore neatly modeled by the DEVS formalism, a discrete-event model of computation, in contrast with traditional methods, which form discrete-time models of the continuous-time system. They have therefore been implemented in [PowerDEVS], a simulation engine for such discrete-event systems.
Theoretical properties
In 2001, Ernesto Kofman proved[2] a remarkable property of the quantized-state system simulation method: namely, that when the technique is used to solve a stable linear time-invariant (LTI) system, the global error is bounded by a constant that is proportional to the quantum, but (crucially) independent of the duration of the simulation. More specifically, for a stable multidimensional LTI system with the state-transition matrix $A$ and input matrix $B$, it was shown in [CK06] that the absolute error vector ${\vec {e}}(t)$ is bounded above by
$\left|{\vec {e}}(t)\right|\leq \left|V\right|\ \left|\Re \left(\Lambda \right)^{-1}\Lambda \right|\ \left|V^{-1}\right|\ \Delta {\vec {Q}}+\left|V\right|\ \left|\Re \left(\Lambda \right)^{-1}V^{-1}B\right|\ \Delta {\vec {u}}$
where $\Delta {\vec {Q}}$ is the vector of state quanta, $\Delta {\vec {u}}$ is the vector with quanta adopted in the input signals, $V\Lambda V^{-1}=A$ is the eigendecomposition or Jordan canonical form of $A$, and $\left|\,\cdot \,\right|$ denotes the element-wise absolute value operator (not to be confused with the determinant or norm).
It is worth noticing that this remarkable error bound comes at a price: the global error for a stable LTI system is also, in a sense, bounded below by the quantum itself, at least for the first-order QSS1 method. This is because, unless the approximation happens to coincide exactly with the correct value (an event which will almost surely not happen), it will simply continue oscillating around the equilibrium, as the state is always (by definition) guaranteed to change by exactly one quantum outside of the equilibrium. Avoiding this condition would require finding a reliable technique for dynamically lowering the quantum in a manner analogous to adaptive stepsize methods in traditional discrete time simulation algorithms.
First-order QSS method – QSS1
Let an initial value problem be specified as follows.
${\dot {x}}(t)=f(x(t),t),\quad x(t_{0})=x_{0}.$
The first-order QSS method, known as QSS1, approximates the above system by
${\dot {x}}(t)=f(q(t),t),\quad q(t_{0})=x_{0}.$
where $x$ and $q$ are related by a hysteretic quantization function
$q(t)={\begin{cases}x(t)&{\text{if }}\left|x(t)-q(t^{-})\right|\geq \Delta Q\\q(t^{-})&{\text{otherwise}}\end{cases}}$
where $\Delta Q$ is called a quantum. Notice that this quantization function is hysteretic because it has memory: not only is its output a function of the current state $x(t)$, but it also depends on its old value, $q(t^{-})$.
This formulation therefore approximates the state by a piecewise constant function, $q(t)$, that updates its value as soon as the state deviates from this approximation by one quantum.
The multidimensional formulation of this system is almost the same as the single-dimensional formulation above: the $k^{\text{th}}$ quantized state $q_{k}(t)$ is a function of its corresponding state, $x_{k}(t)$, and the state vector ${\vec {x}}(t)$ is a function of the entire quantized state vector, ${\vec {q}}(t)$:
${\vec {x}}(t)=f({\vec {q}}(t),t)$
High-order QSS methods – QSS2 and QSS3
The second-order QSS method, QSS2, follows the same principle as QSS1, except that it defines $q(t)$ as a piecewise linear approximation of the trajectory $x(t)$ that updates its trajectory as soon as the two differ from each other by one quantum. The pattern continues for higher-order approximations, which define the quantized state $q(t)$ as successively higher-order polynomial approximations of the system's state.
It is important to note that, while in principle a QSS method of arbitrary order can be used to model a continuous-time system, it is seldom desirable to use methods of order higher than four, as the Abel–Ruffini theorem implies that the time of the next quantization, $t$, cannot (in general) be explicitly solved for algebraically when the polynomial approximation is of degree greater than four, and hence must be approximated iteratively using a root-finding algorithm. In practice, QSS2 or QSS3 proves sufficient for many problems and the use of higher-order methods results in little, if any, additional benefit.
Software implementation
The QSS Methods can be implemented as a discrete event system and simulated in any DEVS simulator.
QSS methods constitute the main numerical solver for PowerDEVS[BK011] software. They have also been implemented in as a stand-alone version.
References
1. Migoni, Gustavo; Ernesto Kofman; François Cellier (2011). "Quantization-based new integration methods for stiff ordinary differential equations". Simulation: 387–407.
2. Kofman, Ernesto (2002). "A second-order approximation for DEVS simulation of continuous systems". Simulation. 78 (2): 76–89. CiteSeerX 10.1.1.640.1903. doi:10.1177/0037549702078002206. S2CID 20959777.
• [CK06] Francois E. Cellier & Ernesto Kofman (2006). Continuous System Simulation (first ed.). Springer. ISBN 978-0-387-26102-7.
• [BK11] Bergero, Federico & Kofman, Ernesto (2011). "PowerDEVS: a tool for hybrid system modeling and real-time simulation" (first ed.). Society for Computer Simulation International,San Diego.
External links
• Stand-alone implementation of QSS Methods
• PowerDEVS at SourceForge
Numerical methods for integration
First-order methods
• Euler method
• Backward Euler
• Semi-implicit Euler
• Exponential Euler
Second-order methods
• Verlet integration
• Velocity Verlet
• Trapezoidal rule
• Beeman's algorithm
• Midpoint method
• Heun's method
• Newmark-beta method
• Leapfrog integration
Higher-order methods
• Exponential integrator
• Runge–Kutta methods
• List of Runge–Kutta methods
• Linear multistep method
• General linear methods
• Backward differentiation formula
• Yoshida
• Gauss–Legendre method
Theory
• Symplectic integrator
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Quantity
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement. Mass, time, distance, heat, and angle are among the familiar examples of quantitative properties.
Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others function as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little.
Under the name of multitude comes what is discontinuous and discrete and divisible ultimately into indivisibles, such as: army, fleet, flock, government, company, party, people, mess (military), chorus, crowd, and number; all which are cases of collective nouns. Under the name of magnitude comes what is continuous and unified and divisible only into smaller divisibles, such as: matter, mass, energy, liquid, material—all cases of non-collective nouns.
Along with analyzing its nature and classification, the issues of quantity involve such closely related topics as dimensionality, equality, proportion, the measurements of quantities, the units of measurements, number and numbering systems, the types of numbers and their relations to each other as numerical ratios.
Background
In mathematics, the concept of quantity is an ancient one extending back to the time of Aristotle and earlier. Aristotle regarded quantity as a fundamental ontological and scientific category. In Aristotle's ontology, quantity or quantum was classified into two different types, which he characterized as follows:
Quantum means that which is divisible into two or more constituent parts, of which each is by nature a one and a this. A quantum is a plurality if it is numerable, a magnitude if it is measurable. Plurality means that which is divisible potentially into non-continuous parts, magnitude that which is divisible into continuous parts; of magnitude, that which is continuous in one dimension is length; in two breadth, in three depth. Of these, limited plurality is number, limited length is a line, breadth a surface, depth a solid.
— Aristotle, Metaphysics, Book V, Ch. 11-14
In his Elements, Euclid developed the theory of ratios of magnitudes without studying the nature of magnitudes, as Archimedes, but giving the following significant definitions:
A magnitude is a part of a magnitude, the less of the greater, when it measures the greater; A ratio is a sort of relation in respect of size between two magnitudes of the same kind.
— Euclid, Elements
For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993). John Wallis later conceived of ratios of magnitudes as real numbers:
When a comparison in terms of ratio is made, the resultant ratio often [namely with the exception of the 'numerical genus' itself] leaves the genus of quantities compared, and passes into the numerical genus, whatever the genus of quantities compared may have been.
— John Wallis, Mathesis Universalis
That is, the ratio of magnitudes of any quantity, whether volume, mass, heat and so on, is a number. Following this, Newton then defined number, and the relationship between quantity and number, in the following terms:
By number we understand not so much a multitude of unities, as the abstracted ratio of any quantity to another quantity of the same kind, which we take for unity.
— Newton, 1728
Structure
Continuous quantities possess a particular structure that was first explicitly characterized by Hölder (1901) as a set of axioms that define such features as identities and relations between magnitudes. In science, quantitative structure is the subject of empirical investigation and cannot be assumed to exist a priori for any given property. The linear continuum represents the prototype of continuous quantitative structure as characterized by Hölder (1901) (translated in Michell & Ernst, 1996). A fundamental feature of any type of quantity is that the relationships of equality or inequality can in principle be stated in comparisons between particular magnitudes, unlike quality, which is marked by likeness, similarity and difference, diversity. Another fundamental feature is additivity. Additivity may involve concatenation, such as adding two lengths A and B to obtain a third A + B. Additivity is not, however, restricted to extensive quantities but may also entail relations between magnitudes that can be established through experiments that permit tests of hypothesized observable manifestations of the additive relations of magnitudes. Another feature is continuity, on which Michell (1999, p. 51) says of length, as a type of quantitative attribute, "what continuity means is that if any arbitrary length, a, is selected as a unit, then for every positive real number, r, there is a length b such that b = ra". A further generalization is given by the theory of conjoint measurement, independently developed by French economist Gérard Debreu (1960) and by the American mathematical psychologist R. Duncan Luce and statistician John Tukey (1964).
In mathematics
Magnitude (how much) and multitude (how many), the two principal types of quantities, are further divided as mathematical and physical. In formal terms, quantities—their ratios, proportions, order and formal relationships of equality and inequality—are studied by mathematics. The essential part of mathematical quantities consists of having a collection of variables, each assuming a set of values. These can be a set of a single quantity, referred to as a scalar when represented by real numbers, or have multiple quantities as do vectors and tensors, two kinds of geometric objects.
The mathematical usage of a quantity can then be varied and so is situationally dependent. Quantities can be used as being infinitesimal, arguments of a function, variables in an expression (independent or dependent), or probabilistic as in random and stochastic quantities. In mathematics, magnitudes and multitudes are also not only two distinct kinds of quantity but furthermore relatable to each other.
Number theory covers the topics of the discrete quantities as numbers: number systems with their kinds and relations. Geometry studies the issues of spatial magnitudes: straight lines, curved lines, surfaces and solids, all with their respective measurements and relationships.
A traditional Aristotelian realist philosophy of mathematics, stemming from Aristotle and remaining popular until the eighteenth century, held that mathematics is the "science of quantity". Quantity was considered to be divided into the discrete (studied by arithmetic) and the continuous (studied by geometry and later calculus). The theory fits reasonably well elementary or school mathematics but less well the abstract topological and algebraic structures of modern mathematics.[1]
In science
Establishing quantitative structure and relationships between different quantities is the cornerstone of modern science, especially but not restricted to physical sciences. Physics is fundamentally a quantitative science; chemistry, biology and others are increasingly so. Their progress is chiefly achieved due to rendering the abstract qualities of material entities into physical quantities, by postulating that all material bodies marked by quantitative properties or physical dimensions are subject to some measurements and observations. Setting the units of measurement, physics covers such fundamental quantities as space (length, breadth, and depth) and time, mass and force, temperature, energy, and quanta.
A distinction has also been made between intensive quantity and extensive quantity as two types of quantitative property, state or relation. The magnitude of an intensive quantity does not depend on the size, or extent, of the object or system of which the quantity is a property, whereas magnitudes of an extensive quantity are additive for parts of an entity or subsystems. Thus, magnitude does depend on the extent of the entity or system in the case of extensive quantity. Examples of intensive quantities are density and pressure, while examples of extensive quantities are energy, volume, and mass.
In natural language
In human languages, including English, number is a syntactic category, along with person and gender. The quantity is expressed by identifiers, definite and indefinite, and quantifiers, definite and indefinite, as well as by three types of nouns: 1. count unit nouns or countables; 2. mass nouns, uncountables, referring to the indefinite, unidentified amounts; 3. nouns of multitude (collective nouns). The word ‘number’ belongs to a noun of multitude standing either for a single entity or for the individuals making the whole. An amount in general is expressed by a special class of words called identifiers, indefinite and definite and quantifiers, definite and indefinite. The amount may be expressed by: singular form and plural from, ordinal numbers before a count noun singular (first, second, third...), the demonstratives; definite and indefinite numbers and measurements (hundred/hundreds, million/millions), or cardinal numbers before count nouns. The set of language quantifiers covers "a few, a great number, many, several (for count names); a bit of, a little, less, a great deal (amount) of, much (for mass names); all, plenty of, a lot of, enough, more, most, some, any, both, each, either, neither, every, no". For the complex case of unidentified amounts, the parts and examples of a mass are indicated with respect to the following: a measure of a mass (two kilos of rice and twenty bottles of milk or ten pieces of paper); a piece or part of a mass (part, element, atom, item, article, drop); or a shape of a container (a basket, box, case, cup, bottle, vessel, jar).
Further examples
Some further examples of quantities are:
• 1.76 litres (liters) of milk, a continuous quantity
• 2πr metres, where r is the length of a radius of a circle expressed in metres (or meters), also a continuous quantity
• one apple, two apples, three apples, where the number is an integer representing the count of a denumerable collection of objects (apples)
• 500 people (also a type of count data)
• a couple conventionally refers to two objects.
• a few usually refers to an indefinite, but usually small number, greater than one.
• quite a few also refers to an indefinite, but surprisingly (in relation to the context) large number.
• several refers to an indefinite, but usually small, number – usually indefinitely greater than "a few".
Dimensionless quantity
This section is an excerpt from Dimensionless quantity.[edit]
A dimensionless quantity (also known as a bare quantity, pure quantity as well as quantity of dimension one)[2] is a quantity to which no physical dimension is assigned. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Dimensionless quantities are distinct from quantities that have associated dimensions, such as time (measured in seconds).
The corresponding unit of measurement is one (symbol 1),[3][4] which is not explicitly shown. For any system of units, the number one is considered a base unit.[5]
Dimensionless units are special names that serve as units of measurement for expressing other dimensionless quantities. For example, in the SI, radians (rad) and steradians (sr) are dimensionless units for plane angles and solid angles, respectively.[3] For example, optical extent is defined as having units of metres multiplied by steradians.[6]
See also
• Quantification (science)
• Observable quantity
• Numerical value equation
References
1. Franklin, James (2014). An Aristotelian Realist Philosophy of Mathematics. Basingstoke: Palgrave Macmillan. p. 31-2. ISBN 9781137400734.
2. "1.8 (1.6) quantity of dimension one dimensionless quantity". International vocabulary of metrology — Basic and general concepts and associated terms (VIM). ISO. 2008. Retrieved 2011-03-22.
3. "SI Brochure: The International System of Units, 9th Edition". BIPM. ISBN 978-92-822-2272-0.
4. Mohr, Peter J.; Phillips, William Daniel (2015-06-01). "Dimensionless units in the SI". Metrologia. 52.
5. "ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23.
6. "17-21-048: optical extent". CIE S 017:2020 ILV: International Lighting Vocabulary, 2nd edition. International Commission on Illumination. Retrieved 2023-02-20.
• Aristotle, Logic (Organon): Categories, in Great Books of the Western World, V.1. ed. by Adler, M.J., Encyclopædia Britannica, Inc., Chicago (1990)
• Aristotle, Physical Treatises: Physics, in Great Books of the Western World, V.1, ed. by Adler, M.J., Encyclopædia Britannica, Inc., Chicago (1990)
• Aristotle, Metaphysics, in Great Books of the Western World, V.1, ed. by Adler, M.J., Encyclopædia Britannica, Inc., Chicago (1990)
• Franklin, J. (2014). Quantity and number, in Neo-Aristotelian Perspectives in Metaphysics, ed. D.D. Novotny and L. Novak, New York: Routledge, 221-44.
• Hölder, O. (1901). Die Axiome der Quantität und die Lehre vom Mass. Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematische-Physicke Klasse, 53, 1-64.
• Klein, J. (1968). Greek Mathematical Thought and the Origin of Algebra. Cambridge. Mass: MIT Press.
• Laycock, H. (2006). Words without Objects: Oxford, Clarendon Press. Oxfordscholarship.com
• Michell, J. (1993). The origins of the representational theory of measurement: Helmholtz, Hölder, and Russell. Studies in History and Philosophy of Science, 24, 185-206.
• Michell, J. (1999). Measurement in Psychology. Cambridge: Cambridge University Press.
• Michell, J. & Ernst, C. (1996). The axioms of quantity and the theory of measurement: translated from Part I of Otto Hölder's German text "Die Axiome der Quantität und die Lehre vom Mass". Journal of Mathematical Psychology, 40, 235-252.
• Newton, I. (1728/1967). Universal Arithmetic: Or, a Treatise of Arithmetical Composition and Resolution. In D.T. Whiteside (Ed.), The mathematical Works of Isaac Newton, Vol. 2 (pp. 3–134). New York: Johnson Reprint Corp.
• Wallis, J. Mathesis universalis (as quoted in Klein, 1968).
External links
Look up quantity or few in Wiktionary, the free dictionary.
Wikiquote has quotations related to Quantity.
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Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles.[2]: 1.1 It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.
Part of a series of articles about
Quantum mechanics
$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle $
Schrödinger equation
• Introduction
• Glossary
• History
Background
• Classical mechanics
• Old quantum theory
• Bra–ket notation
• Hamiltonian
• Interference
Fundamentals
• Complementarity
• Decoherence
• Entanglement
• Energy level
• Measurement
• Nonlocality
• Quantum number
• State
• Superposition
• Symmetry
• Tunnelling
• Uncertainty
• Wave function
• Collapse
Experiments
• Bell's inequality
• Davisson–Germer
• Double-slit
• Elitzur–Vaidman
• Franck–Hertz
• Leggett–Garg inequality
• Mach–Zehnder
• Popper
• Quantum eraser
• Delayed-choice
• Schrödinger's cat
• Stern–Gerlach
• Wheeler's delayed-choice
Formulations
• Overview
• Heisenberg
• Interaction
• Matrix
• Phase-space
• Schrödinger
• Sum-over-histories (path integral)
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
Advanced topics
• Relativistic quantum mechanics
• Quantum field theory
• Quantum information science
• Quantum computing
• Quantum chaos
• EPR paradox
• Density matrix
• Scattering theory
• Quantum statistical mechanics
• Quantum machine learning
Scientists
• Aharonov
• Bell
• Bethe
• Blackett
• Bloch
• Bohm
• Bohr
• Born
• Bose
• de Broglie
• Compton
• Dirac
• Davisson
• Debye
• Ehrenfest
• Einstein
• Everett
• Fock
• Fermi
• Feynman
• Glauber
• Gutzwiller
• Heisenberg
• Hilbert
• Jordan
• Kramers
• Pauli
• Lamb
• Landau
• Laue
• Moseley
• Millikan
• Onnes
• Planck
• Rabi
• Raman
• Rydberg
• Schrödinger
• Simmons
• Sommerfeld
• von Neumann
• Weyl
• Wien
• Wigner
• Zeeman
• Zeilinger
Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.[3]
Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).
Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper, which explained the photoelectric effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born, Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle's energy, momentum, and other physical properties may yield.
Overview and fundamental concepts
Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms,[4] but its application to human beings raises philosophical problems, such as Wigner's friend, and its application to the universe as a whole remains speculative.[5] Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy.[note 1]
A fundamental feature of the theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number, known as a probability amplitude. This is known as the Born rule, named after physicist Max Born. For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another.
One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between different measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum.
Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment. In the basic version of this experiment, a coherent light source, such as a laser beam, illuminates a plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate.[6]: 102–111 [2]: 1.1–1.8 The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles.[6] However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave).[6]: 109 [7][8] However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. Other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit.[2] This behavior is known as wave–particle duality.
Another counter-intuitive phenomenon predicted by quantum mechanics is quantum tunnelling: a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential.[9] In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay, nuclear fusion in stars, and applications such as scanning tunnelling microscopy and the tunnel diode.[10]
When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "...the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought".[11] Quantum entanglement enables the counter-intuitive properties of quantum pseudo-telepathy, and can be a valuable resource in communication protocols, such as quantum key distribution and superdense coding.[12] Contrary to popular misconception, entanglement does not allow sending signals faster than light, as demonstrated by the no-communication theorem.[12]
Another possibility opened by entanglement is testing for "hidden variables", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory can provide. A collection of results, most significantly Bell's theorem, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed, using entangled particles, and they have shown results incompatible with the constraints imposed by local hidden variables.[13][14]
It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra, differential equations, group theory, and other more advanced subjects.[note 2] Accordingly, this article will present a mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples.
Mathematical formulation
Main article: Mathematical formulation of quantum mechanics
In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector $\psi $ belonging to a (separable) complex Hilbert space ${\mathcal {H}}$. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys $\langle \psi ,\psi \rangle =1$, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, $\psi $ and $e^{i\alpha }\psi $ represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions $L^{2}(\mathbb {C} )$, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors $\mathbb {C} ^{2}$ with the usual inner product.
Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue $\lambda $ is non-degenerate and the probability is given by $|\langle {\vec {\lambda }},\psi \rangle |^{2}$, where ${\vec {\lambda }}$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by $\langle \psi ,P_{\lambda }\psi \rangle $, where $P_{\lambda }$ is the projector onto its associated eigenspace. In the continuous case, these formulas give instead the probability density.
After the measurement, if result $\lambda $ was obtained, the quantum state is postulated to collapse to ${\vec {\lambda }}$, in the non-degenerate case, or to $P_{\lambda }\psi /{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}$, in the general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wave function collapse" (see, for example, the many-worlds interpretation). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.[17]
The time evolution of a quantum state is described by the Schrödinger equation:
$i\hbar {\frac {d}{dt}}\psi (t)=H\psi (t).$
Here $H$ denotes the Hamiltonian, the observable corresponding to the total energy of the system, and $\hbar $ is the reduced Planck constant. The constant $i\hbar $ is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle.
The solution of this differential equation is given by
$\psi (t)=e^{-iHt/\hbar }\psi (0).$
The operator $U(t)=e^{-iHt/\hbar }$ is known as the time-evolution operator, and has the crucial property that it is unitary. This time evolution is deterministic in the sense that – given an initial quantum state $\psi (0)$ – it makes a definite prediction of what the quantum state $\psi (t)$ will be at any later time.[18]
Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital (Fig. 1).
Analytic solutions of the Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom. Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment.
However, there are techniques for finding approximate solutions. One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy. Another method is called "semi-classical equation of motion", which applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos.
Uncertainty principle
One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[19][20] Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator ${\hat {X}}$ and momentum operator ${\hat {P}}$ do not commute, but rather satisfy the canonical commutation relation:
$[{\hat {X}},{\hat {P}}]=i\hbar .$
Given a quantum state, the Born rule lets us compute expectation values for both $X$ and $P$, and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation, we have
$\sigma _{X}={\sqrt {\langle {X}^{2}\rangle -\langle {X}\rangle ^{2}}},$
and likewise for the momentum:
$\sigma _{P}={\sqrt {\langle {P}^{2}\rangle -\langle {P}\rangle ^{2}}}.$
The uncertainty principle states that
$\sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}}.$
Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[21] This inequality generalizes to arbitrary pairs of self-adjoint operators $A$ and $B$. The commutator of these two operators is
$[A,B]=AB-BA,$
and this provides the lower bound on the product of standard deviations:
$\sigma _{A}\sigma _{B}\geq {\frac {1}{2}}\left|\langle [A,B]\rangle \right|.$
Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an $i/\hbar $ factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space. This is why in quantum equations in position space, the momentum $p_{i}$ is replaced by $-i\hbar {\frac {\partial }{\partial x}}$, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times $-\hbar ^{2}$.[19]
Composite systems and entanglement
When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces ${\mathcal {H}}_{A}$ and ${\mathcal {H}}_{B}$, respectively. The Hilbert space of the composite system is then
${\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}.$
If the state for the first system is the vector $\psi _{A}$ and the state for the second system is $\psi _{B}$, then the state of the composite system is
$\psi _{A}\otimes \psi _{B}.$
Not all states in the joint Hilbert space ${\mathcal {H}}_{AB}$ can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if $\psi _{A}$ and $\phi _{A}$ are both possible states for system $A$, and likewise $\psi _{B}$ and $\phi _{B}$ are both possible states for system $B$, then
${\tfrac {1}{\sqrt {2}}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right)$
is a valid joint state that is not separable. States that are not separable are called entangled.[22][23]
If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[22][23] Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[22][24]
As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.[25]
Equivalence between formulations
There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).[26] An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics.
Symmetries and conservation laws
Main article: Noether's theorem
The Hamiltonian $H$ is known as the generator of time evolution, since it defines a unitary time-evolution operator $U(t)=e^{-iHt/\hbar }$ for each value of $t$. From this relation between $U(t)$ and $H$, it follows that any observable $A$ that commutes with $H$ will be conserved: its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator $A$ can generate a family of unitary operators parameterized by a variable $t$. Under the evolution generated by $A$, any observable $B$ that commutes with $A$ will be conserved. Moreover, if $B$ is conserved by evolution under $A$, then $A$ is conserved under the evolution generated by $B$. This implies a quantum version of the result proven by Emmy Noether in classical (Lagrangian) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law.
Examples
Free particle
The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:
$H={\frac {1}{2m}}P^{2}=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}.$
The general solution of the Schrödinger equation is given by
$\psi (x,t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {\psi }}(k,0)e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}\mathrm {d} k,$
which is a superposition of all possible plane waves $e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}$, which are eigenstates of the momentum operator with momentum $p=\hbar k$. The coefficients of the superposition are ${\hat {\psi }}(k,0)$, which is the Fourier transform of the initial quantum state $\psi (x,0)$.
It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states.[note 3] Instead, we can consider a Gaussian wave packet:
$\psi (x,0)={\frac {1}{\sqrt[{4}]{\pi a}}}e^{-{\frac {x^{2}}{2a}}}$
which has Fourier transform, and therefore momentum distribution
${\hat {\psi }}(k,0)={\sqrt[{4}]{\frac {a}{\pi }}}e^{-{\frac {ak^{2}}{2}}}.$
We see that as we make $a$ smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making $a$ larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.
As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.[27]
Particle in a box
The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region.[19]: 77–78 For the one-dimensional case in the $x$ direction, the time-independent Schrödinger equation may be written
$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\psi }{dx^{2}}}=E\psi .$
With the differential operator defined by
${\hat {p}}_{x}=-i\hbar {\frac {d}{dx}}$
the previous equation is evocative of the classic kinetic energy analogue,
${\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,$
with state $\psi $ in this case having energy $E$ coincident with the kinetic energy of the particle.
The general solutions of the Schrödinger equation for the particle in a box are
$\psi (x)=Ae^{ikx}+Be^{-ikx}\qquad \qquad E={\frac {\hbar ^{2}k^{2}}{2m}}$
or, from Euler's formula,
$\psi (x)=C\sin(kx)+D\cos(kx).\!$
The infinite potential walls of the box determine the values of $C,D,$ and $k$ at $x=0$ and $x=L$ where $\psi $ must be zero. Thus, at $x=0$,
$\psi (0)=0=C\sin(0)+D\cos(0)=D$
and $D=0$. At $x=L$,
$\psi (L)=0=C\sin(kL),$
in which $C$ cannot be zero as this would conflict with the postulate that $\psi $ has norm 1. Therefore, since $\sin(kL)=0$, $kL$ must be an integer multiple of $\pi $,
$k={\frac {n\pi }{L}}\qquad \qquad n=1,2,3,\ldots .$
This constraint on $k$ implies a constraint on the energy levels, yielding
$E_{n}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac {n^{2}h^{2}}{8mL^{2}}}.$
A finite potential well is the generalization of the infinite potential well problem to potential wells having finite depth. The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of the well. Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well. Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance of modern technologies such as flash memory and scanning tunneling microscopy.
Harmonic oscillator
As in the classical case, the potential for the quantum harmonic oscillator is given by
$V(x)={\frac {1}{2}}m\omega ^{2}x^{2}.$
This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by
$\psi _{n}(x)={\sqrt {\frac {1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad $
$n=0,1,2,\ldots .$
where Hn are the Hermite polynomials
$H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),$
and the corresponding energy levels are
$E_{n}=\hbar \omega \left(n+{1 \over 2}\right).$
This is another example illustrating the discretization of energy for bound states.
Mach–Zehnder interferometer
The Mach–Zehnder interferometer (MZI) illustrates the concepts of superposition and interference with linear algebra in dimension 2, rather than differential equations. It can be seen as a simplified version of the double-slit experiment, but it is of interest in its own right, for example in the delayed choice quantum eraser, the Elitzur–Vaidman bomb tester, and in studies of quantum entanglement.[28][29]
We can model a photon going through the interferometer by considering that at each point it can be in a superposition of only two paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right. The quantum state of the photon is therefore a vector $\psi \in \mathbb {C} ^{2}$ that is a superposition of the "lower" path $\psi _{l}={\begin{pmatrix}1\\0\end{pmatrix}}$ and the "upper" path $\psi _{u}={\begin{pmatrix}0\\1\end{pmatrix}}$, that is, $\psi =\alpha \psi _{l}+\beta \psi _{u}$ for complex $\alpha ,\beta $. In order to respect the postulate that $\langle \psi ,\psi \rangle =1$ we require that $|\alpha |^{2}+|\beta |^{2}=1$.
Both beam splitters are modelled as the unitary matrix $B={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&i\\i&1\end{pmatrix}}$, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of $1/{\sqrt {2}}$, or be reflected to the other path with a probability amplitude of $i/{\sqrt {2}}$. The phase shifter on the upper arm is modelled as the unitary matrix $P={\begin{pmatrix}1&0\\0&e^{i\Delta \Phi }\end{pmatrix}}$, which means that if the photon is on the "upper" path it will gain a relative phase of $\Delta \Phi $, and it will stay unchanged if it is in the lower path.
A photon that enters the interferometer from the left will then be acted upon with a beam splitter $B$, a phase shifter $P$, and another beam splitter $B$, and so end up in the state
$BPB\psi _{l}=ie^{i\Delta \Phi /2}{\begin{pmatrix}-\sin(\Delta \Phi /2)\\\cos(\Delta \Phi /2)\end{pmatrix}},$
and the probabilities that it will be detected at the right or at the top are given respectively by
$p(u)=|\langle \psi _{u},BPB\psi _{l}\rangle |^{2}=\cos ^{2}{\frac {\Delta \Phi }{2}},$
$p(l)=|\langle \psi _{l},BPB\psi _{l}\rangle |^{2}=\sin ^{2}{\frac {\Delta \Phi }{2}}.$
One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities.
It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters. This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired). In both cases, there will be no interference between the paths anymore, and the probabilities are given by $p(u)=p(l)=1/2$, independently of the phase $\Delta \Phi $. From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it is in a genuine quantum superposition of the two paths.[30]
Applications
Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by classical methods.[note 4] Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particles that make up all forms of matter (electrons, protons, neutrons, photons, and others). Solid-state physics and materials science are dependent upon quantum mechanics.[31]
In many aspects, modern technology operates at a scale where quantum effects are significant. Important applications of quantum theory include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the optical amplifier and the laser, the transistor and semiconductors such as the microprocessor, medical and research imaging such as magnetic resonance imaging and electron microscopy.[32] Explanations for many biological and physical phenomena are rooted in the nature of the chemical bond, most notably the macro-molecule DNA.
Relation to other scientific theories
Modern physics
${\hat {H}}|\psi _{n}(t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi _{n}(t)\rangle $
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Schrödinger and Einstein field equations
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Classical mechanics
The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators. These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions. An important guide for making these choices is the correspondence principle, a heuristic which states that the predictions of quantum mechanics reduce to those of classical mechanics in the regime of large quantum numbers.[33] One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical model in the correspondence limit. This approach is known as quantization.
When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.
Complications arise with chaotic systems, which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in these systems.
Quantum decoherence is a mechanism through which quantum systems lose coherence, and thus become incapable of displaying many typically quantum effects: quantum superpositions become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations. Quantum coherence is not typically evident at macroscopic scales, except maybe at temperatures approaching absolute zero at which quantum behavior may manifest macroscopically.[note 5]
Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts. For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical, thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.[34]
Special relativity and electrodynamics
Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein–Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles). The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.[35][36]
The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical $\textstyle -e^{2}/(4\pi \epsilon _{_{0}}r)$ Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.
Quantum field theories for the strong nuclear force and the weak nuclear force have also been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of subnuclear particles such as quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory (known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.[37]
Relation to general relativity
Even though the predictions of both quantum theory and general relativity have been supported by rigorous and repeated empirical evidence, their abstract formalisms contradict each other and they have proven extremely difficult to incorporate into one consistent, cohesive model. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those particular applications. However, the lack of a correct theory of quantum gravity is an important issue in physical cosmology and the search by physicists for an elegant "Theory of Everything" (TOE). Consequently, resolving the inconsistencies between both theories has been a major goal of 20th- and 21st-century physics. This TOE would combine not only the models of subatomic physics but also derive the four fundamental forces of nature from a single force or phenomenon.[38]
One proposal for doing so is string theory, which posits that the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force.[39][40]
Another popular theory is loop quantum gravity (LQG), which describes quantum properties of gravity and is thus a theory of quantum spacetime. LQG is an attempt to merge and adapt standard quantum mechanics and standard general relativity. This theory describes space as an extremely fine fabric "woven" of finite loops called spin networks. The evolution of a spin network over time is called a spin foam. The characteristic length scale of a spin foam is the Planck length, approximately 1.616×10−35 m, and so lengths shorter than the Planck length are not physically meaningful in LQG.[41]
Philosophical implications
Unsolved problem in physics:
Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the "superposition of states" and "wave function collapse", give rise to the reality we perceive?
(more unsolved problems in physics)
Since its inception, the many counter-intuitive aspects and results of quantum mechanics have provoked strong philosophical debates and many interpretations. The arguments centre on the probabilistic nature of quantum mechanics, the difficulties with wavefunction collapse and the related measurement problem, and quantum nonlocality. Perhaps the only consensus that exists about these issues is that there is no consensus. Richard Feynman once said, "I think I can safely say that nobody understands quantum mechanics."[42] According to Steven Weinberg, "There is now in my opinion no entirely satisfactory interpretation of quantum mechanics."[43]
The views of Niels Bohr, Werner Heisenberg and other physicists are often grouped together as the "Copenhagen interpretation".[44][45] According to these views, the probabilistic nature of quantum mechanics is not a temporary feature which will eventually be replaced by a deterministic theory, but is instead a final renunciation of the classical idea of "causality". Bohr in particular emphasized that any well-defined application of the quantum mechanical formalism must always make reference to the experimental arrangement, due to the complementary nature of evidence obtained under different experimental situations. Copenhagen-type interpretations were adopted by Nobel laureates in quantum physics, including Bohr,[46] Heisenberg,[47] Schrodinger,[48], Feynman[2], and Zeilinger[49] as well as 21st century researchers in quantum foundations.[50]
Albert Einstein, himself one of the founders of quantum theory, was troubled by its apparent failure to respect some cherished metaphysical principles, such as determinism and locality. Einstein's long-running exchanges with Bohr about the meaning and status of quantum mechanics are now known as the Bohr–Einstein debates. Einstein believed that underlying quantum mechanics must be a theory that explicitly forbids action at a distance. He argued that quantum mechanics was incomplete, a theory that was valid but not fundamental, analogous to how thermodynamics is valid, but the fundamental theory behind it is statistical mechanics. In 1935, Einstein and his collaborators Boris Podolsky and Nathan Rosen published an argument that the principle of locality implies the incompleteness of quantum mechanics, a thought experiment later termed the Einstein–Podolsky–Rosen paradox.[note 6] In 1964, John Bell showed that EPR's principle of locality, together with determinism, was actually incompatible with quantum mechanics: they implied constraints on the correlations produced by distance systems, now known as Bell inequalities, that can be violated by entangled particles.[55] Since then several experiments have been performed to obtain these correlations, with the result that they do in fact violate Bell inequalities, and thus falsify the conjunction of locality with determinism.[13][14]
Bohmian mechanics shows that it is possible to reformulate quantum mechanics to make it deterministic, at the price of making it explicitly nonlocal. It attributes not only a wave function to a physical system, but in addition a real position, that evolves deterministically under a nonlocal guiding equation. The evolution of a physical system is given at all times by the Schrödinger equation together with the guiding equation; there is never a collapse of the wave function. This solves the measurement problem.[56]
Everett's many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a multiverse composed of mostly independent parallel universes.[57] This is a consequence of removing the axiom of the collapse of the wave packet. All possible states of the measured system and the measuring apparatus, together with the observer, are present in a real physical quantum superposition. While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we do not observe the multiverse as a whole, but only one parallel universe at a time. Exactly how this is supposed to work has been the subject of much debate. Several attempts have been made to make sense of this and derive the Born rule,[58][59] with no consensus on whether they have been successful.[60][61][62]
Relational quantum mechanics appeared in the late 1990s as a modern derivative of Copenhagen-type ideas,[63] and QBism was developed some years later.[64]
History
Quantum mechanics was developed in the early decades of the 20th century, driven by the need to explain phenomena that, in some cases, had been observed in earlier times. Scientific inquiry into the wave nature of light began in the 17th and 18th centuries, when scientists such as Robert Hooke, Christiaan Huygens and Leonhard Euler proposed a wave theory of light based on experimental observations.[65] In 1803 English polymath Thomas Young described the famous double-slit experiment.[66] This experiment played a major role in the general acceptance of the wave theory of light.
During the early 19th century, chemical research by John Dalton and Amedeo Avogadro lent weight to the atomic theory of matter, an idea that James Clerk Maxwell, Ludwig Boltzmann and others built upon to establish the kinetic theory of gases. The successes of kinetic theory gave further credence to the idea that matter is composed of atoms, yet the theory also had shortcomings that would only be resolved by the development of quantum mechanics.[67] While the early conception of atoms from Greek philosophy had been that they were indivisible units – the word "atom" deriving from the Greek for "uncuttable" – the 19th century saw the formulation of hypotheses about subatomic structure. One important discovery in that regard was Michael Faraday's 1838 observation of a glow caused by an electrical discharge inside a glass tube containing gas at low pressure. Julius Plücker, Johann Wilhelm Hittorf and Eugen Goldstein carried on and improved upon Faraday's work, leading to the identification of cathode rays, which J. J. Thomson found to consist of subatomic particles that would be called electrons.[68][69]
The black-body radiation problem was discovered by Gustav Kirchhoff in 1859. In 1900, Max Planck proposed the hypothesis that energy is radiated and absorbed in discrete "quanta" (or energy packets), yielding a calculation that precisely matched the observed patterns of black-body radiation.[70] The word quantum derives from the Latin, meaning "how great" or "how much".[71] According to Planck, quantities of energy could be thought of as divided into "elements" whose size (E) would be proportional to their frequency (ν):
$E=h\nu \ $,
where h is Planck's constant. Planck cautiously insisted that this was only an aspect of the processes of absorption and emission of radiation and was not the physical reality of the radiation.[72] In fact, he considered his quantum hypothesis a mathematical trick to get the right answer rather than a sizable discovery.[73] However, in 1905 Albert Einstein interpreted Planck's quantum hypothesis realistically and used it to explain the photoelectric effect, in which shining light on certain materials can eject electrons from the material. Niels Bohr then developed Planck's ideas about radiation into a model of the hydrogen atom that successfully predicted the spectral lines of hydrogen.[74] Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle (later called the photon), with a discrete amount of energy that depends on its frequency.[75] In his paper "On the Quantum Theory of Radiation," Einstein expanded on the interaction between energy and matter to explain the absorption and emission of energy by atoms. Although overshadowed at the time by his general theory of relativity, this paper articulated the mechanism underlying the stimulated emission of radiation,[76] which became the basis of the laser.
This phase is known as the old quantum theory. Never complete or self-consistent, the old quantum theory was rather a set of heuristic corrections to classical mechanics.[77] The theory is now understood as a semi-classical approximation[78] to modern quantum mechanics.[79] Notable results from this period include, in addition to the work of Planck, Einstein and Bohr mentioned above, Einstein and Peter Debye's work on the specific heat of solids, Bohr and Hendrika Johanna van Leeuwen's proof that classical physics cannot account for diamagnetism, and Arnold Sommerfeld's extension of the Bohr model to include special-relativistic effects.
In the mid-1920s quantum mechanics was developed to become the standard formulation for atomic physics. In 1923, the French physicist Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa. Building on de Broglie's approach, modern quantum mechanics was born in 1925, when the German physicists Werner Heisenberg, Max Born, and Pascual Jordan[80][81] developed matrix mechanics and the Austrian physicist Erwin Schrödinger invented wave mechanics. Born introduced the probabilistic interpretation of Schrödinger's wave function in July 1926.[82] Thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927.[83]
By 1930 quantum mechanics had been further unified and formalized by David Hilbert, Paul Dirac and John von Neumann[84] with greater emphasis on measurement, the statistical nature of our knowledge of reality, and philosophical speculation about the 'observer'. It has since permeated many disciplines, including quantum chemistry, quantum electronics, quantum optics, and quantum information science. It also provides a useful framework for many features of the modern periodic table of elements, and describes the behaviors of atoms during chemical bonding and the flow of electrons in computer semiconductors, and therefore plays a crucial role in many modern technologies. While quantum mechanics was constructed to describe the world of the very small, it is also needed to explain some macroscopic phenomena such as superconductors[85] and superfluids.[86]
See also
• Bra–ket notation
• Einstein's thought experiments
• List of textbooks on classical and quantum mechanics
• Macroscopic quantum phenomena
• Phase-space formulation
• Regularization (physics)
• Two-state quantum system
Explanatory notes
1. See, for example, Precision tests of QED. The relativistic refinement of quantum mechanics known as quantum electrodynamics (QED) has been shown to agree with experiment to within 1 part in 108 for some atomic properties.
2. Physicist John C. Baez cautions, "there's no way to understand the interpretation of quantum mechanics without also being able to solve quantum mechanics problems – to understand the theory, you need to be able to use it (and vice versa)".[15] Carl Sagan outlined the "mathematical underpinning" of quantum mechanics and wrote, "For most physics students, this might occupy them from, say, third grade to early graduate school – roughly 15 years. [...] The job of the popularizer of science, trying to get across some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason."[16]
3. A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes introduce fictitious "bases" for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.[19]: 100–105
4. See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 14–11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II, pp. 8–6), and lasers (vol III, pp. 9–13).
5. see macroscopic quantum phenomena, Bose–Einstein condensate, and Quantum machine
6. The published form of the EPR argument was due to Podolsky, and Einstein himself was not satisfied with it. In his own publications and correspondence, Einstein used a different argument to insist that quantum mechanics is an incomplete theory.[51][52][53][54]
References
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Further reading
The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.
• Chester, Marvin (1987). Primer of Quantum Mechanics. John Wiley. ISBN 0-486-42878-8
• Cox, Brian; Forshaw, Jeff (2011). The Quantum Universe: Everything That Can Happen Does Happen. Allen Lane. ISBN 978-1-84614-432-5.
• Richard Feynman, 1985. QED: The Strange Theory of Light and Matter, Princeton University Press. ISBN 0-691-08388-6. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing many insights for the expert.
• Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most technical of the works cited here. Passages using algebra, trigonometry, and bra–ket notation can be passed over on a first reading.
• N. David Mermin, 1990, "Spooky actions at a distance: mysteries of the QT" in his Boojums All the Way Through. Cambridge University Press: 110–76.
• Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo, NY: Prometheus Books. Chpts. 5–8. Includes cosmological and philosophical considerations.
More technical:
• Bernstein, Jeremy (2009). Quantum Leaps. Cambridge, Massachusetts: Belknap Press of Harvard University Press. ISBN 978-0-674-03541-6.
• Bohm, David (1989). Quantum Theory. Dover Publications. ISBN 978-0-486-65969-5.
• Binney, James; Skinner, David (2008). The Physics of Quantum Mechanics. Oxford University Press. ISBN 978-0-19-968857-9.
• Eisberg, Robert; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). Wiley. ISBN 978-0-471-87373-0.
• Bryce DeWitt, R. Neill Graham, eds., 1973. The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press. ISBN 0-691-08131-X
• Everett, Hugh (1957). "Relative State Formulation of Quantum Mechanics". Reviews of Modern Physics. 29 (3): 454–462. Bibcode:1957RvMP...29..454E. doi:10.1103/RevModPhys.29.454. S2CID 17178479.
• Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. Vol. 1–3. Addison-Wesley. ISBN 978-0-7382-0008-8.
• D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. Compendium of quantum physics, Concepts, experiments, history and philosophy, Springer-Verlag, Berlin, Heidelberg.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-111892-8. OCLC 40251748. A standard undergraduate text.
• Max Jammer, 1966. The Conceptual Development of Quantum Mechanics. McGraw Hill.
• Hagen Kleinert, 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. Singapore: World Scientific. Draft of 4th edition. Archived 2008-06-15 at the Wayback Machine
• L.D. Landau, E.M. Lifshitz (1977). Quantum Mechanics: Non-Relativistic Theory. Vol. 3 (3rd ed.). Pergamon Press. ISBN 978-0-08-020940-1. Online copy
• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 978-0-8053-8714-8.
• Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon Press. ISBN 0-08-203204-1
• George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0-486-43517-2.
• Merzbacher, Eugen (1998). Quantum Mechanics. Wiley, John & Sons, Inc. ISBN 978-0-471-88702-7.
• Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G.M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III. online
• Omnès, Roland (1999). Understanding Quantum Mechanics. Princeton University Press. ISBN 978-0-691-00435-8. OCLC 39849482.
• Scerri, Eric R., 2006. The Periodic Table: Its Story and Its Significance. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. ISBN 0-19-530573-6
• Shankar, R. (1994). Principles of Quantum Mechanics. Springer. ISBN 978-0-306-44790-7.
• Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press. ISBN 978-0-691-13968-5.
• Transnational College of Lex (1996). What is Quantum Mechanics? A Physics Adventure. Language Research Foundation, Boston. ISBN 978-0-9643504-1-0. OCLC 34661512.
• Veltman, Martinus J.G. (2003), Facts and Mysteries in Elementary Particle Physics.
On Wikibooks
• This Quantum World
External links
• J. O'Connor and E. F. Robertson: A history of quantum mechanics.
• Introduction to Quantum Theory at Quantiki.
• Quantum Physics Made Relatively Simple: three video lectures by Hans Bethe
Course material
• Quantum Cook Book and PHYS 201: Fundamentals of Physics II by Ramamurti Shankar, Yale OpenCourseware
• Modern Physics: With waves, thermodynamics, and optics – an online textbook.
• MIT OpenCourseWare: Chemistry and Physics. See 8.04, 8.05 and 8.06
• 5½ Examples in Quantum Mechanics
• Imperial College Quantum Mechanics Course.
Philosophy
• Ismael, Jenann. "Quantum Mechanics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Krips, Henry. "Measurement in Quantum Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
Quantum mechanics
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Quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics.[1][2][3] In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved.[2] QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.[2][3]
Quantum field theory
Feynman diagram
History
Background
• Field theory
• Electromagnetism
• Weak force
• Strong force
• Quantum mechanics
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• Gauge theory
• Yang–Mills theory
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• Symmetry in quantum mechanics
• C-symmetry
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• Explicit symmetry breaking
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• Noether charge
• Topological charge
Tools
• Anomaly
• Background field method
• BRST quantization
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• Quantum electrodynamics
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In technical terms, QED can be described as a very accurate way to calculate the probability of the position and movement of particles, even those massless such as photons, and the quantity depending on position (field) of those particles, and described light and matter beyond the wave-particle duality proposed by Einstein in 1905. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen.[2]: Ch1
History
The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac, who (during the 1920s) was able to compute the coefficient of spontaneous emission of an atom.[4]
Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and an elegant formulation of quantum electrodynamics by Enrico Fermi,[5] physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck,[6] and Victor Weisskopf,[7] in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer.[8] At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics.
Difficulties with the theory increased through the end of the 1940s. Improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom,[9] now known as the Lamb shift and magnetic moment of the electron.[10] These experiments exposed discrepancies which the theory was unable to explain.
A first indication of a possible way out was given by Hans Bethe in 1947,[11] after attending the Shelter Island Conference.[12] While he was traveling by train from the conference to Schenectady he made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford.[11] Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named renormalization.
Based on Bethe's intuition and fundamental papers on the subject by Shin'ichirō Tomonaga,[13] Julian Schwinger,[14][15] Richard Feynman[1][16][17] and Freeman Dyson,[18][19] it was finally possible to get fully covariant formulations that were finite at any order in a perturbation series of quantum electrodynamics. Shin'ichirō Tomonaga, Julian Schwinger and Richard Feynman were jointly awarded with the 1965 Nobel Prize in Physics for their work in this area.[20] Their contributions, and those of Freeman Dyson, were about covariant and gauge-invariant formulations of quantum electrodynamics that allow computations of observables at any order of perturbation theory. Feynman's mathematical technique, based on his diagrams, initially seemed very different from the field-theoretic, operator-based approach of Schwinger and Tomonaga, but Freeman Dyson later showed that the two approaches were equivalent.[18] Renormalization, the need to attach a physical meaning at certain divergences appearing in the theory through integrals, has subsequently become one of the fundamental aspects of quantum field theory and has come to be seen as a criterion for a theory's general acceptability. Even though renormalization works very well in practice, Feynman was never entirely comfortable with its mathematical validity, even referring to renormalization as a "shell game" and "hocus pocus".[2]: 128
Thence, neither Feynman nor Dirac were happy with that way to approach the observations made in theoretical physics, above all in quantum mechanics.[21]
QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory is quantum chromodynamics, which began in the early 1960s and attained its present form in the 1970s work by H. David Politzer, Sidney Coleman, David Gross and Frank Wilczek. Building on the pioneering work of Schwinger, Gerald Guralnik, Dick Hagen, and Tom Kibble,[22][23] Peter Higgs, Jeffrey Goldstone, and others, Sheldon Glashow, Steven Weinberg and Abdus Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.
Feynman's view of quantum electrodynamics
Introduction
Near the end of his life, Richard Feynman gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), QED: The Strange Theory of Light and Matter,[2] a classic non-mathematical exposition of QED from the point of view articulated below.
The key components of Feynman's presentation of QED are three basic actions.[2]: 85
A photon goes from one place and time to another place and time.
An electron goes from one place and time to another place and time.
An electron emits or absorbs a photon at a certain place and time.
These actions are represented in the form of visual shorthand by the three basic elements of Feynman diagrams: a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram.
As well as the visual shorthand for the actions, Feynman introduces another kind of shorthand for the numerical quantities called probability amplitudes. The probability is the square of the absolute value of total probability amplitude, ${\text{probability}}=|f({\text{amplitude}})|^{2}$. If a photon moves from one place and time $A$ to another place and time $B$, the associated quantity is written in Feynman's shorthand as $P(A{\text{ to }}B)$, and it depends on only the momentum and polarization of the photon. The similar quantity for an electron moving from $C$ to $D$ is written $E(C{\text{ to }}D)$. It depends on the momentum and polarization of the electron, in addition to a constant Feynman calls n, sometimes called the "bare" mass of the electron: it is related to, but not the same as, the measured electron mass. Finally, the quantity that tells us about the probability amplitude for an electron to emit or absorb a photon Feynman calls j, and is sometimes called the "bare" charge of the electron: it is a constant, and is related to, but not the same as, the measured electron charge e.[2]: 91
QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above (P(A to B), E(C to D) and j) acts just like our everyday probability (a simplification made in Feynman's book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.
The basic rules of probability amplitudes that will be used are:[2]: 93
1. If an event can occur via a number of indistinguishable alternative processes (a.k.a. "virtual" processes), then its probability amplitude is the sum of the probability amplitudes of the alternatives.
2. If a virtual process involves a number of independent or concomitant sub-processes, then the probability amplitude of the total (compound) process is the product of the probability amplitudes of the sub-processes.
The indistinguishability criterion in (a) is very important: it means that there is no observable feature present in the given system that in any way "reveals" which alternative is taken. In such a case, one cannot observe which alternative actually takes place without changing the experimental setup in some way (e.g. by introducing a new apparatus into the system). Whenever one is able to observe which alternative takes place, one always finds that the probability of the event is the sum of the probabilities of the alternatives. Indeed, if this were not the case, the very term "alternatives" to describe these processes would be inappropriate. What (a) says is that once the physical means for observing which alternative occurred is removed, one cannot still say that the event is occurring through "exactly one of the alternatives" in the sense of adding probabilities; one must add the amplitudes instead.[2]: 82
Similarly, the independence criterion in (b) is very important: it only applies to processes which are not "entangled".
Basic constructions
Suppose we start with one electron at a certain place and time (this place and time being given the arbitrary label A) and a photon at another place and time (given the label B). A typical question from a physical standpoint is: "What is the probability of finding an electron at C (another place and a later time) and a photon at D (yet another place and time)?". The simplest process to achieve this end is for the electron to move from A to C (an elementary action) and for the photon to move from B to D (another elementary action). From a knowledge of the probability amplitudes of each of these sub-processes – E(A to C) and P(B to D) – we would expect to calculate the probability amplitude of both happening together by multiplying them, using rule b) above. This gives a simple estimated overall probability amplitude, which is squared to give an estimated probability.
But there are other ways in which the result could come about. The electron might move to a place and time E, where it absorbs the photon; then move on before emitting another photon at F; then move on to C, where it is detected, while the new photon moves on to D. The probability of this complex process can again be calculated by knowing the probability amplitudes of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption. We would expect to find the total probability amplitude by multiplying the probability amplitudes of each of the actions, for any chosen positions of E and F. We then, using rule a) above, have to add up all these probability amplitudes for all the alternatives for E and F. (This is not elementary in practice and involves integration.) But there is another possibility, which is that the electron first moves to G, where it emits a photon, which goes on to D, while the electron moves on to H, where it absorbs the first photon, before moving on to C. Again, we can calculate the probability amplitude of these possibilities (for all points G and H). We then have a better estimation for the total probability amplitude by adding the probability amplitudes of these two possibilities to our original simple estimate. Incidentally, the name given to this process of a photon interacting with an electron in this way is Compton scattering.
There is an infinite number of other intermediate "virtual" processes in which more and more photons are absorbed and/or emitted. For each of these processes, a Feynman diagram could be drawn describing it. This implies a complex computation for the resulting probability amplitudes, but provided it is the case that the more complicated the diagram, the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of any interactive process between electrons and photons, it is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability amplitude.
That basic scaffolding remains when one moves to a quantum description, but some conceptual changes are needed. One is that whereas we might expect in our everyday life that there would be some constraints on the points to which a particle can move, that is not true in full quantum electrodynamics. There is a nonzero probability amplitude of an electron at A, or a photon at B, moving as a basic action to any other place and time in the universe. That includes places that could only be reached at speeds greater than that of light and also earlier times. (An electron moving backwards in time can be viewed as a positron moving forward in time.)[2]: 89, 98–99
Probability amplitudes
Quantum mechanics introduces an important change in the way probabilities are computed. Probabilities are still represented by the usual real numbers we use for probabilities in our everyday world, but probabilities are computed as the square modulus of probability amplitudes, which are complex numbers.
Feynman avoids exposing the reader to the mathematics of complex numbers by using a simple but accurate representation of them as arrows on a piece of paper or screen. (These must not be confused with the arrows of Feynman diagrams, which are simplified representations in two dimensions of a relationship between points in three dimensions of space and one of time.) The amplitude arrows are fundamental to the description of the world given by quantum theory. They are related to our everyday ideas of probability by the simple rule that the probability of an event is the square of the length of the corresponding amplitude arrow. So, for a given process, if two probability amplitudes, v and w, are involved, the probability of the process will be given either by
$P=|\mathbf {v} +\mathbf {w} |^{2}$
or
$P=|\mathbf {v} \,\mathbf {w} |^{2}.$
The rules as regards adding or multiplying, however, are the same as above. But where you would expect to add or multiply probabilities, instead you add or multiply probability amplitudes that now are complex numbers.
Addition and multiplication are common operations in the theory of complex numbers and are given in the figures. The sum is found as follows. Let the start of the second arrow be at the end of the first. The sum is then a third arrow that goes directly from the beginning of the first to the end of the second. The product of two arrows is an arrow whose length is the product of the two lengths. The direction of the product is found by adding the angles that each of the two have been turned through relative to a reference direction: that gives the angle that the product is turned relative to the reference direction.
That change, from probabilities to probability amplitudes, complicates the mathematics without changing the basic approach. But that change is still not quite enough because it fails to take into account the fact that both photons and electrons can be polarized, which is to say that their orientations in space and time have to be taken into account. Therefore, P(A to B) consists of 16 complex numbers, or probability amplitude arrows.[2]: 120–121 There are also some minor changes to do with the quantity j, which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping.
Associated with the fact that the electron can be polarized is another small necessary detail, which is connected with the fact that an electron is a fermion and obeys Fermi–Dirac statistics. The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include (as we always must) the complementary Feynman diagram in which we exchange two electron events, the resulting amplitude is the reverse – the negative – of the first. The simplest case would be two electrons starting at A and B ending at C and D. The amplitude would be calculated as the "difference", E(A to D) × E(B to C) − E(A to C) × E(B to D), where we would expect, from our everyday idea of probabilities, that it would be a sum.[2]: 112–113
Propagators
Finally, one has to compute P(A to B) and E(C to D) corresponding to the probability amplitudes for the photon and the electron respectively. These are essentially the solutions of the Dirac equation, which describe the behavior of the electron's probability amplitude and the Maxwell's equations, which describes the behavior of the photon's probability amplitude. These are called Feynman propagators. The translation to a notation commonly used in the standard literature is as follows:
$P(A{\text{ to }}B)\to D_{F}(x_{B}-x_{A}),\quad E(C{\text{ to }}D)\to S_{F}(x_{D}-x_{C}),$
where a shorthand symbol such as $x_{A}$ stands for the four real numbers that give the time and position in three dimensions of the point labeled A.
Mass renormalization
A problem arose historically which held up progress for twenty years: although we start with the assumption of three basic "simple" actions, the rules of the game say that if we want to calculate the probability amplitude for an electron to get from A to B, we must take into account all the possible ways: all possible Feynman diagrams with those endpoints. Thus there will be a way in which the electron travels to C, emits a photon there and then absorbs it again at D before moving on to B. Or it could do this kind of thing twice, or more. In short, we have a fractal-like situation in which if we look closely at a line, it breaks up into a collection of "simple" lines, each of which, if looked at closely, are in turn composed of "simple" lines, and so on ad infinitum. This is a challenging situation to handle. If adding that detail only altered things slightly, then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to infinite probability amplitudes. In time this problem was "fixed" by the technique of renormalization. However, Feynman himself remained unhappy about it, calling it a "dippy process",[2]: 128 and Dirac also criticized this procedure as "in mathematics one does not get rid of infinities when it does not please you".[21]
Conclusions
Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as the anomalous magnetic dipole moment. However, as Feynman points out, it fails to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem."[2]: 152
Mathematical formulation
QED action
Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action[24]: 78
QED Action
$S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi \right]$
where
• $\gamma ^{\mu }$ are Dirac matrices.
• $\psi $ a bispinor field of spin-1/2 particles (e.g. electron–positron field).
• ${\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}$, called "psi-bar", is sometimes referred to as the Dirac adjoint.
• $D_{\mu }\equiv \partial _{\mu }+ieA_{\mu }+ieB_{\mu }$ is the gauge covariant derivative.
• e is the coupling constant, equal to the electric charge of the bispinor field.
• $A_{\mu }$ is the covariant four-potential of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a ${\text{U}}(1)$ connection.
• $B_{\mu }$ is the external field imposed by external source.
• m is the mass of the electron or positron.
• $F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ is the electromagnetic field tensor. This is also known as the curvature of the gauge field.
Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field $B_{\mu }$ set to zero for simplicity)
${\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi -ej^{\mu }A_{\mu }$
where $j^{\mu }$ is the conserved ${\text{U}}(1)$ current arising from Noether's theorem. It is written
$j^{\mu }={\bar {\psi }}\gamma ^{\mu }\psi .$
Equations of motion
Expanding the covariant derivative in the Lagrangian gives
${\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -e{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi -m{\bar {\psi }}\psi $
$=-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+i{\bar {\psi }}\gamma ^{\mu }\partial _{\mu }\psi -m{\bar {\psi }}\psi -ej^{\mu }A_{\mu }.$
For simplicity, $B_{\mu }$ has been set to zero. Alternatively, we can absorb $B_{\mu }$ into a new gauge field $A'_{\mu }=A_{\mu }+B_{\mu }$ and relabel the new field as $A_{\mu }.$
From this Lagrangian, the equations of motion for the $\psi $ and $A_{\mu }$ fields can be obtained.
Equation of motion for ψ
These arise most straightforwardly by considering the Euler-Lagrange equation for ${\bar {\psi }}$. Since the Lagrangian contains no $\partial _{\mu }{\bar {\psi }}$ terms, we immediately get
${\frac {\partial {\mathcal {L}}}{\partial {\bar {\psi }}}}=0$
so the equation of motion can be written $(i\gamma ^{\mu }\partial _{\mu }-m)\psi =e\gamma ^{\mu }A_{\mu }\psi .$
Equation of motion for Aμ
• Using the Euler–Lagrange equation for the $A_{\mu }$ field,
$\partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)-{\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=0,$
(3)
the derivatives this time are
$\partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)=\partial _{\nu }\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right),$
${\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=-e{\bar {\psi }}\gamma ^{\mu }\psi .$
Substituting back into (3) leads to
$\partial _{\mu }F^{\mu \nu }=e{\bar {\psi }}\gamma ^{\nu }\psi $
which can be written in terms of the ${\text{U}}(1)$ current $j^{\mu }$ as
$\partial _{\mu }F^{\mu \nu }=ej^{\nu }.$
Now, if we impose the Lorenz gauge condition
$\partial _{\mu }A^{\mu }=0,$
the equations reduce to
$\Box A^{\mu }=ej^{\mu },$
which is a wave equation for the four-potential, the QED version of the classical Maxwell equations in the Lorenz gauge. (The square represents the wave operator, $\Box =\partial _{\mu }\partial ^{\mu }$.)
Interaction picture
This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator, which for a given initial state $|i\rangle $ will give a final state $\langle f|$ in such a way to have[24]: 5
$M_{fi}=\langle f|U|i\rangle .$
This technique is also known as the S-matrix. The evolution operator is obtained in the interaction picture, where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:[24]: 123
$V=e\int d^{3}x\,{\bar {\psi }}\gamma ^{\mu }\psi A_{\mu },$
and so, one has[24]: 86
$U=T\exp \left[-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'\,V(t')\right],$
where T is the time-ordering operator. This evolution operator only has meaning as a series, and what we get here is a perturbation series with the fine-structure constant as the development parameter. This series is called the Dyson series.
Feynman diagrams
Despite the conceptual clarity of this Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations, it is much easier to work with the Fourier transforms of the propagators. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles' momenta rather than their positions are considered, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then look the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.
Using Wick's theorem on the terms of the Dyson series, all the terms of the S-matrix for quantum electrodynamics can be computed through the technique of Feynman diagrams. In this case, rules for drawing are the following[24]: 801–802
To these rules we must add a further one for closed loops that implies an integration on momenta $ \int d^{4}p/(2\pi )^{4}$, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see Propagator for details). The signature of the metric $\eta _{\mu \nu }$ is ${\rm {diag}}(+---)$.
From them, computations of probability amplitudes are straightforwardly given. An example is Compton scattering, with an electron and a photon undergoing elastic scattering. Feynman diagrams are in this case[24]: 158–159
and so we are able to get the corresponding amplitude at the first order of a perturbation series for the S-matrix:
$M_{fi}=(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}{\frac {p\!\!\!/+k\!\!\!/+m_{e}}{(p+k)^{2}-m_{e}^{2}}}\epsilon \!\!\!/({\vec {k}},\lambda )u({\vec {p}},s)+(ie)^{2}{\overline {u}}({\vec {p}}',s')\epsilon \!\!\!/({\vec {k}},\lambda ){\frac {p\!\!\!/-k\!\!\!/'+m_{e}}{(p-k')^{2}-m_{e}^{2}}}\epsilon \!\!\!/\,'({\vec {k}}',\lambda ')^{*}u({\vec {p}},s),$
from which we can compute the cross section for this scattering.
Nonperturbative phenomena
The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams. However, quantum electrodynamics also leads to predictions beyond perturbation theory. In the presence of very strong electric fields, it predicts that electrons and positrons will be spontaneously produced, so causing the decay of the field. This process, called the Schwinger effect,[25] cannot be understood in terms of any finite number of Feynman diagrams and hence is described as nonperturbative. Mathematically, it can be derived by a semiclassical approximation to the path integral of quantum electrodynamics.
Renormalizability
Higher-order terms can be straightforwardly computed for the evolution operator, but these terms display diagrams containing the following simpler ones[24]: ch 10
• One-loop contribution to the vacuum polarization function $\Pi $
• One-loop contribution to the electron self-energy function $\Sigma $
• One-loop contribution to the vertex function $\Gamma $
that, being closed loops, imply the presence of diverging integrals having no mathematical meaning. To overcome this difficulty, a technique called renormalization has been devised, producing finite results in very close agreement with experiments. A criterion for the theory being meaningful after renormalization is that the number of diverging diagrams is finite. In this case, the theory is said to be "renormalizable". The reason for this is that to get observables renormalized, one needs a finite number of constants to maintain the predictive value of the theory untouched. This is exactly the case of quantum electrodynamics displaying just three diverging diagrams. This procedure gives observables in very close agreement with experiment as seen e.g. for electron gyromagnetic ratio.
Renormalizability has become an essential criterion for a quantum field theory to be considered as a viable one. All the theories describing fundamental interactions, except gravitation, whose quantum counterpart is only conjectural and presently under very active research, are renormalizable theories.
Nonconvergence of series
An argument by Freeman Dyson shows that the radius of convergence of the perturbation series in QED is zero.[26] The basic argument goes as follows: if the coupling constant were negative, this would be equivalent to the Coulomb force constant being negative. This would "reverse" the electromagnetic interaction so that like charges would attract and unlike charges would repel. This would render the vacuum unstable against decay into a cluster of electrons on one side of the universe and a cluster of positrons on the other side of the universe. Because the theory is "sick" for any negative value of the coupling constant, the series does not converge but is at best an asymptotic series.
From a modern perspective, we say that QED is not well defined as a quantum field theory to arbitrarily high energy.[27] The coupling constant runs to infinity at finite energy, signalling a Landau pole. The problem is essentially that QED appears to suffer from quantum triviality issues. This is one of the motivations for embedding QED within a Grand Unified Theory.
Electrodynamics in curved spacetime
See also: Maxwell's equations in curved spacetime and Dirac equation in curved spacetime
This theory can be extended, at least as a classical field theory, to curved spacetime. This arises similarly to the flat spacetime case, from coupling a free electromagnetic theory to a free fermion theory and including an interaction which promotes the partial derivative in the fermion theory to a gauge-covariant derivative.
See also
• Abraham–Lorentz force
• Anomalous magnetic moment
• Bhabha scattering
• Cavity quantum electrodynamics
• Circuit quantum electrodynamics
• Compton scattering
• Euler–Heisenberg Lagrangian
• Gupta–Bleuler formalism
• Lamb shift
• Landau pole
• Moeller scattering
• Non-relativistic quantum electrodynamics
• Photon polarization
• Positronium
• Precision tests of QED
• QED vacuum
• QED: The Strange Theory of Light and Matter
• Quantization of the electromagnetic field
• Scalar electrodynamics
• Schrödinger equation
• Schwinger model
• Schwinger–Dyson equation
• Vacuum polarization
• Vertex function
• Wheeler–Feynman absorber theory
References
1. R. P. Feynman (1949). "Space–Time Approach to Quantum Electrodynamics". Physical Review. 76 (6): 769–89. Bibcode:1949PhRv...76..769F. doi:10.1103/PhysRev.76.769.
2. Feynman, Richard (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6.
3. Feynman, R. P. (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction". Physical Review. 80 (3): 440–457. Bibcode:1950PhRv...80..440F. doi:10.1103/PhysRev.80.440.
4. P. A. M. Dirac (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proceedings of the Royal Society of London A. 114 (767): 243–65. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039.
5. E. Fermi (1932). "Quantum Theory of Radiation". Reviews of Modern Physics. 4 (1): 87–132. Bibcode:1932RvMP....4...87F. doi:10.1103/RevModPhys.4.87.
6. Bloch, F.; Nordsieck, A. (1937). "Note on the Radiation Field of the Electron". Physical Review. 52 (2): 54–59. Bibcode:1937PhRv...52...54B. doi:10.1103/PhysRev.52.54.
7. V. F. Weisskopf (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". Physical Review. 56 (1): 72–85. Bibcode:1939PhRv...56...72W. doi:10.1103/PhysRev.56.72.
8. R. Oppenheimer (1930). "Note on the Theory of the Interaction of Field and Matter". Physical Review. 35 (5): 461–77. Bibcode:1930PhRv...35..461O. doi:10.1103/PhysRev.35.461.
9. Lamb, Willis; Retherford, Robert (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method". Physical Review. 72 (3): 241–43. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241.
10. Foley, H.M.; Kusch, P. (1948). "On the Intrinsic Moment of the Electron". Physical Review. 73 (3): 412. Bibcode:1948PhRv...73..412F. doi:10.1103/PhysRev.73.412.
11. H. Bethe (1947). "The Electromagnetic Shift of Energy Levels". Physical Review. 72 (4): 339–41. Bibcode:1947PhRv...72..339B. doi:10.1103/PhysRev.72.339. S2CID 120434909.
12. Schweber, Silvan (1994). "Chapter 5". QED and the Men Who Did it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press. p. 230. ISBN 978-0-691-03327-3.
13. S. Tomonaga (1946). "On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields". Progress of Theoretical Physics. 1 (2): 27–42. Bibcode:1946PThPh...1...27T. doi:10.1143/PTP.1.27.
14. J. Schwinger (1948). "On Quantum-Electrodynamics and the Magnetic Moment of the Electron". Physical Review. 73 (4): 416–17. Bibcode:1948PhRv...73..416S. doi:10.1103/PhysRev.73.416.
15. J. Schwinger (1948). "Quantum Electrodynamics. I. A Covariant Formulation". Physical Review. 74 (10): 1439–61. Bibcode:1948PhRv...74.1439S. doi:10.1103/PhysRev.74.1439.
16. R. P. Feynman (1949). "The Theory of Positrons". Physical Review. 76 (6): 749–59. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749. S2CID 120117564.
17. R. P. Feynman (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction" (PDF). Physical Review. 80 (3): 440–57. Bibcode:1950PhRv...80..440F. doi:10.1103/PhysRev.80.440.
18. F. Dyson (1949). "The Radiation Theories of Tomonaga, Schwinger, and Feynman". Physical Review. 75 (3): 486–502. Bibcode:1949PhRv...75..486D. doi:10.1103/PhysRev.75.486.
19. F. Dyson (1949). "The S Matrix in Quantum Electrodynamics". Physical Review. 75 (11): 1736–55. Bibcode:1949PhRv...75.1736D. doi:10.1103/PhysRev.75.1736.
20. "The Nobel Prize in Physics 1965". Nobel Foundation. Retrieved 2008-10-09.
21. The story of the positron - Paul Dirac (1975), retrieved 2023-07-19
22. Guralnik, G. S.; Hagen, C. R.; Kibble, T. W. B. (1964). "Global Conservation Laws and Massless Particles". Physical Review Letters. 13 (20): 585–87. Bibcode:1964PhRvL..13..585G. doi:10.1103/PhysRevLett.13.585.
23. Guralnik, G. S. (2009). "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles". International Journal of Modern Physics A. 24 (14): 2601–27. arXiv:0907.3466. Bibcode:2009IJMPA..24.2601G. doi:10.1142/S0217751X09045431. S2CID 16298371.
24. Peskin, Michael; Schroeder, Daniel (1995). An introduction to quantum field theory (Reprint ed.). Westview Press. ISBN 978-0201503975.
25. Schwinger, Julian (1951-06-01). "On Gauge Invariance and Vacuum Polarization". Physical Review. American Physical Society (APS). 82 (5): 664–679. Bibcode:1951PhRv...82..664S. doi:10.1103/physrev.82.664. ISSN 0031-899X.
26. Kinoshita, Toichiro (June 5, 1997). "Quantum Electrodynamics has Zero Radius of Convergence Summarized from Toichiro Kinoshita". Retrieved May 6, 2017.
27. Espriu and Tarrach (Apr 30, 1996). "Ambiguities in QED: Renormalons versus Triviality". Physics Letters B. 383 (4): 482–486. arXiv:hep-ph/9604431. Bibcode:1996PhLB..383..482E. doi:10.1016/0370-2693(96)00779-4. S2CID 119095192.
Further reading
Books
• Berestetskii, V. B.; Lifshitz, E. M.; Pitaevskii, L. P. (1982). Course of Theoretical Physics, Volume 4: Quantum Electrodynamics (2 ed.). Elsevier. ISBN 978-0-7506-3371-0.
• De Broglie, L. (1925). Recherches sur la theorie des quanta [Research on quantum theory]. France: Wiley-Interscience.
• Feynman, R. P. (1998). Quantum Electrodynamics (New ed.). Westview Press. ISBN 978-0-201-36075-2.
• Greiner, W.; Bromley, D. A.; Müller, B. (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0.
• Jauch, J. M.; Rohrlich, F. (1980). The Theory of Photons and Electrons. Springer-Verlag. ISBN 978-0-387-07295-1.
• Kane, G. L. (1993). Modern Elementary Particle Physics. Westview Press. ISBN 978-0-201-62460-1.
• Miller, A. I. (1995). Early Quantum Electrodynamics: A Sourcebook. Cambridge University Press. ISBN 978-0-521-56891-3.
• Milonni, P. W. (1994). The Quantum Vacuum: An Introduction to Quantum Electrodynamics. Boston: Academic Press. ISBN 0124980805. LCCN 93029780. OCLC 422797902.
• Schweber, S. S. (1994). QED and the Men Who Made It. Princeton University Press. ISBN 978-0-691-03327-3.
• Schwinger, J. (1958). Selected Papers on Quantum Electrodynamics. Dover Publications. ISBN 978-0-486-60444-2.
• Tannoudji-Cohen, C.; Dupont-Roc, Jacques; Grynberg, Gilbert (1997). Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley-Interscience. ISBN 978-0-471-18433-1.
Journals
• Dudley, J.M.; Kwan, A.M. (1996). "Richard Feynman's popular lectures on quantum electrodynamics: The 1979 Robb Lectures at Auckland University". American Journal of Physics. 64 (6): 694–98. Bibcode:1996AmJPh..64..694D. doi:10.1119/1.18234.
External links
• Feynman's Nobel Prize lecture describing the evolution of QED and his role in it
• Feynman's New Zealand lectures on QED for non-physicists
• http://qed.wikina.org/ – Animations demonstrating QED
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Quantum Fourier transform
In quantum computing, the quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem. The quantum Fourier transform was discovered by Don Coppersmith.[1]
The quantum Fourier transform can be performed efficiently on a quantum computer with a decomposition into the product of simpler unitary matrices. The discrete Fourier transform on $2^{n}$ amplitudes can be implemented as a quantum circuit consisting of only $O(n^{2})$ Hadamard gates and controlled phase shift gates, where $n$ is the number of qubits.[2] This can be compared with the classical discrete Fourier transform, which takes $O(n2^{n})$ gates (where $n$ is the number of bits), which is exponentially more than $O(n^{2})$.
The quantum Fourier transform acts on a quantum state vector (a quantum register), and the classical Fourier transform acts on a vector. Both types of vectors can be written as lists of complex numbers. In the quantum case it is a sequence of probability amplitudes for all the possible outcomes upon measurement (called basis states, or eigenstates). Because measurement collapses the quantum state to a single basis state, not every task that uses the classical Fourier transform can take advantage of the quantum Fourier transform's exponential speedup.
The best quantum Fourier transform algorithms known (as of late 2000) require only $O(n\log n)$ gates to achieve an efficient approximation.[3]
Definition
The quantum Fourier transform is the classical discrete Fourier transform applied to the vector of amplitudes of a quantum state, which usually has length $N=2^{n}$.
The classical Fourier transform acts on a vector $(x_{0},x_{1},\ldots ,x_{N-1})\in \mathbb {C} ^{N}$ and maps it to the vector $(y_{0},y_{1},\ldots ,y_{N-1})\in \mathbb {C} ^{N}$ according to the formula:
$y_{k}={\frac {1}{\sqrt {N}}}\sum _{n=0}^{N-1}x_{n}\omega _{N}^{-nk},\quad k=0,1,2,\ldots ,N-1,$
where $\omega _{N}=e^{\frac {2\pi i}{N}}$ and $\omega _{N}^{n}$ is an N-th root of unity.
Similarly, the quantum Fourier transform acts on a quantum state $ |x\rangle =\sum _{i=0}^{N-1}x_{i}|i\rangle $ and maps it to a quantum state $ \sum _{i=0}^{N-1}y_{i}|i\rangle $ according to the formula:
$y_{k}={\frac {1}{\sqrt {N}}}\sum _{n=0}^{N-1}x_{n}\omega _{N}^{nk},\quad k=0,1,2,\ldots ,N-1,$
(Conventions for the sign of the phase factor exponent vary; here the quantum Fourier transform has the same effect as the inverse discrete Fourier transform, and vice versa.)
Since $\omega _{N}^{n}$ is a rotation, the inverse quantum Fourier transform acts similarly but with:
$x_{n}={\frac {1}{\sqrt {N}}}\sum _{k=0}^{N-1}y_{k}\omega _{N}^{-nk},\quad n=0,1,2,\ldots ,N-1,$
In case that $|x\rangle $ is a basis state, the quantum Fourier Transform can also be expressed as the map
${\text{QFT}}:|x\rangle \mapsto {\frac {1}{\sqrt {N}}}\sum _{k=0}^{N-1}\omega _{N}^{xk}|k\rangle .$
Equivalently, the quantum Fourier transform can be viewed as a unitary matrix (or quantum gate) acting on quantum state vectors, where the unitary matrix $F_{N}$ is the DFT matrix
$F_{N}={\frac {1}{\sqrt {N}}}{\begin{bmatrix}1&1&1&1&\cdots &1\\1&\omega &\omega ^{2}&\omega ^{3}&\cdots &\omega ^{N-1}\\1&\omega ^{2}&\omega ^{4}&\omega ^{6}&\cdots &\omega ^{2(N-1)}\\1&\omega ^{3}&\omega ^{6}&\omega ^{9}&\cdots &\omega ^{3(N-1)}\\\vdots &\vdots &\vdots &\vdots &&\vdots \\1&\omega ^{N-1}&\omega ^{2(N-1)}&\omega ^{3(N-1)}&\cdots &\omega ^{(N-1)(N-1)}\end{bmatrix}}$
where $\omega =\omega _{N}$. For example, in the case of $N=4=2^{2}$ and phase $\omega =i$ the transformation matrix is
$F_{4}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&i&-1&-i\\1&-1&1&-1\\1&-i&-1&i\end{bmatrix}}$
See also: Generalizations of Pauli matrices § Construction: The clock and shift matrices
Properties
Unitarity
Most of the properties of the quantum Fourier transform follow from the fact that it is a unitary transformation. This can be checked by performing matrix multiplication and ensuring that the relation $FF^{\dagger }=F^{\dagger }F=I$ holds, where $F^{\dagger }$ is the Hermitian adjoint of $F$. Alternately, one can check that orthogonal vectors of norm 1 get mapped to orthogonal vectors of norm 1.
From the unitary property it follows that the inverse of the quantum Fourier transform is the Hermitian adjoint of the Fourier matrix, thus $F^{-1}=F^{\dagger }$. Since there is an efficient quantum circuit implementing the quantum Fourier transform, the circuit can be run in reverse to perform the inverse quantum Fourier transform. Thus both transforms can be efficiently performed on a quantum computer.
Circuit implementation
The quantum gates used in the circuit of $n$ qubits are the Hadamard gate and the phase gate $R_{n}$, here in terms of $N=2^{n}$
$H={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}\qquad {\text{and}}\qquad R_{n}={\begin{pmatrix}1&0\\0&\omega _{N}\end{pmatrix}}$
with $\omega _{N}=e^{i2\pi /N}$ the $N$-th root of unity. The circuit is composed of $H$ gates and the controlled version of $R_{n}$
An orthonormal basis consists of the basis states
$|0\rangle ,\ldots ,|2^{n}-1\rangle .$
These basis states span all possible states of the qubits:
$|x\rangle =|x_{1}x_{2}\ldots x_{n}\rangle =|x_{1}\rangle \otimes |x_{2}\rangle \otimes \cdots \otimes |x_{n}\rangle $
where, with tensor product notation $\otimes $, $|x_{j}\rangle $ indicates that qubit $j$ is in state $x_{j}$, with $x_{j}$ either 0 or 1. By convention, the basis state index $x$ is the binary number encoded by the $x_{j}$, with $x_{1}$ the most significant bit. The quantum Fourier transform can be written as the tensor product of a series of terms:
${\text{QFT}}(|x\rangle )={\frac {1}{\sqrt {N}}}\bigotimes _{j=1}^{n}\left(|0\rangle +\omega _{N}^{x2^{n-j}}|1\rangle \right).$
Using the fractional binary notation
$[0.x_{1}\ldots x_{m}]=\sum _{k=1}^{m}x_{k}2^{-k}.$
the action of the quantum Fourier transform can be expressed in a compact manner:
${\text{QFT}}(|x_{1}x_{2}\ldots x_{n}\rangle )={\frac {1}{\sqrt {N}}}\ \left(|0\rangle +e^{2\pi i\,[0.x_{n}]}|1\rangle \right)\otimes \left(|0\rangle +e^{2\pi i\,[0.x_{n-1}x_{n}]}|1\rangle \right)\otimes \cdots \otimes \left(|0\rangle +e^{2\pi i\,[0.x_{1}x_{2}\ldots x_{n}]}|1\rangle \right).$
To obtain this state from the circuit depicted above, a swap operation of the qubits must be performed to reverse their order. At most $n/2$ swaps are required.[2]
Because the discrete Fourier transform, an operation on n qubits, can be factored into the tensor product of n single-qubit operations, it is easily represented as a quantum circuit (up to an order reversal of the output). Each of those single-qubit operations can be implemented efficiently using one Hadamard gate and a linear number of controlled phase gates. The first term requires one Hadamard gate and $(n-1)$ controlled phase gates, the next term requires one Hadamard gate and $(n-2)$ controlled phase gate, and each following term requires one fewer controlled phase gate. Summing up the number of gates, excluding the ones needed for the output reversal, gives $n+(n-1)+\cdots +1=n(n+1)/2=O(n^{2})$ gates, which is quadratic polynomial in the number of qubits.
Example
The quantum Fourier transform on three qubits, $F_{8}$ with $n=3,N=8=2^{3}$, is represented by the following transformation:
${\text{QFT}}:|x\rangle \mapsto {\frac {1}{\sqrt {8}}}\sum _{k=0}^{7}\omega ^{xk}|k\rangle ,$
where $\omega =\omega _{8}$ is an eighth root of unity satisfying $\omega ^{8}=\left(e^{\frac {i2\pi }{8}}\right)^{8}=1$.
The matrix representation of the Fourier transform on three qubits is:
$F_{8}={\frac {1}{\sqrt {8}}}{\begin{bmatrix}1&1&1&1&1&1&1&1\\1&\omega &\omega ^{2}&\omega ^{3}&\omega ^{4}&\omega ^{5}&\omega ^{6}&\omega ^{7}\\1&\omega ^{2}&\omega ^{4}&\omega ^{6}&1&\omega ^{2}&\omega ^{4}&\omega ^{6}\\1&\omega ^{3}&\omega ^{6}&\omega &\omega ^{4}&\omega ^{7}&\omega ^{2}&\omega ^{5}\\1&\omega ^{4}&1&\omega ^{4}&1&\omega ^{4}&1&\omega ^{4}\\1&\omega ^{5}&\omega ^{2}&\omega ^{7}&\omega ^{4}&\omega &\omega ^{6}&\omega ^{3}\\1&\omega ^{6}&\omega ^{4}&\omega ^{2}&1&\omega ^{6}&\omega ^{4}&\omega ^{2}\\1&\omega ^{7}&\omega ^{6}&\omega ^{5}&\omega ^{4}&\omega ^{3}&\omega ^{2}&\omega \\\end{bmatrix}}.$
The 3-qubit quantum Fourier transform can be rewritten as:
${\text{QFT}}(|x_{1},x_{2},x_{3}\rangle )={\frac {1}{\sqrt {8}}}\ \left(|0\rangle +e^{2\pi i\,[0.x_{3}]}|1\rangle \right)\otimes \left(|0\rangle +e^{2\pi i\,[0.x_{2}x_{3}]}|1\rangle \right)\otimes \left(|0\rangle +e^{2\pi i\,[0.x_{1}x_{2}x_{3}]}|1\rangle \right).$
The following sketch shows the respective circuit for $n=3$ (with reversed order of output qubits with respect to the proper QFT):
As calculated above, the number of gates used is $n(n+1)/2$ which is equal to $6$, for $n=3$.
Relation to quantum Hadamard transform
Using the generalized Fourier transform on finite (abelian) groups, there are actually two natural ways to define a quantum Fourier transform on an n-qubit quantum register. The QFT as defined above is equivalent to the DFT, which considers these n qubits as indexed by the cyclic group $\mathbb {Z} /2^{n}\mathbb {Z} $. However, it also makes sense to consider the qubits as indexed by the Boolean group $(\mathbb {Z} /2\mathbb {Z} )^{n}$, and in this case the Fourier transform is the Hadamard transform. This is achieved by applying a Hadamard gate to each of the n qubits in parallel.[4][5] Shor's algorithm uses both types of Fourier transforms, an initial Hadamard transform as well as a QFT.
References
1. Coppersmith, D. (1994). "An approximate Fourier transform useful in quantum factoring". Technical Report RC19642, IBM. arXiv:quant-ph/0201067.
2. Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9. OCLC 174527496.
3. Hales, L.; Hallgren, S. (November 12–14, 2000). "An improved quantum Fourier transform algorithm and applications". Proceedings 41st Annual Symposium on Foundations of Computer Science. pp. 515–525. CiteSeerX 10.1.1.29.4161. doi:10.1109/SFCS.2000.892139. ISBN 0-7695-0850-2. S2CID 424297.
4. Fourier Analysis of Boolean Maps– A Tutorial –, pp. 12-13
5. Lecture 5: Basic quantum algorithms, Rajat Mittal, pp. 4-5
• K. R. Parthasarathy, Lectures on Quantum Computation and Quantum Error Correcting Codes (Indian Statistical Institute, Delhi Center, June 2001)
• John Preskill, Lecture Notes for Physics 229: Quantum Information and Computation (CIT, September 1998)
External links
• Wolfram Demonstration Project: Quantum Circuit Implementing Grover's Search Algorithm
• Wolfram Demonstration Project: Quantum Circuit Implementing Quantum Fourier Transform
• Quirk online life quantum fourier transform
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
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Wikipedia
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QIP (complexity)
In computational complexity theory, the class QIP (which stands for Quantum Interactive Polynomial time) is the quantum computing analogue of the classical complexity class IP, which is the set of problems solvable by an interactive proof system with a polynomial-time verifier and one computationally unbounded prover. Informally, IP is the set of languages for which a computationally unbounded prover can convince a polynomial-time verifier to accept when the input is in the language (with high probability) and cannot convince the verifier to accept when the input is not in the language (again, with high probability). In other words, the prover and verifier may interact for polynomially many rounds, and if the input is in the language the verifier should accept with probability greater than 2/3, and if the input is not in the language, the verifier should be reject with probability greater than 2/3. In IP, the verifier is like a BPP machine. In QIP, the communication between the prover and verifier is quantum, and the verifier can perform quantum computation. In this case the verifier is like a BQP machine.
By restricting the number of messages used in the protocol to at most k, we get the complexity class QIP(k). QIP and QIP(k) were introduced by John Watrous,[1] who along with Kitaev proved in a later paper[2] that QIP = QIP(3), which shows that 3 messages are sufficient to simulate a polynomial-round quantum interactive protocol. Since QIP(3) is already QIP, this leaves 4 possibly different classes: QIP(0), which is BQP, QIP(1), which is QMA, QIP(2) and QIP.
Kitaev and Watrous also showed that QIP is contained in EXP, the class of problems solvable by a deterministic Turing machine in exponential time.[2] QIP(2) was then shown to be contained in PSPACE, the set of problems solvable by a deterministic Turing machine in polynomial space.[3] Both results were subsumed by the 2009 result that QIP is contained in PSPACE,[4] which also proves that QIP = IP = PSPACE, since PSPACE is easily shown to be in QIP using the result IP = PSPACE.
References
1. Watrous, John (2003), "PSPACE has constant-round quantum interactive proof systems", Theor. Comput. Sci., Essex, UK: Elsevier Science Publishers Ltd., 292 (3): 575–588, doi:10.1016/S0304-3975(01)00375-9, ISSN 0304-3975
2. Kitaev, Alexei; Watrous, John (2000), "Parallelization, amplification, and exponential time simulation of quantum interactive proof systems", STOC '00: Proceedings of the thirty-second annual ACM symposium on Theory of computing, ACM, pp. 608–617, ISBN 978-1-58113-184-0
3. Jain, Rahul; Upadhyay, Sarvagya; Watrous, John (2009), "Two-Message Quantum Interactive Proofs Are in PSPACE", FOCS '09: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, pp. 534–543, ISBN 978-0-7695-3850-1
4. Jain, Rahul; Ji, Zhengfeng; Upadhyay, Sarvagya; Watrous, John (2010), "QIP = PSPACE", STOC '10: Proceedings of the 42nd ACM symposium on Theory of computing, ACM, pp. 573–582, ISBN 978-1-4503-0050-6
External links
• Complexity Zoo: QIP
Quantum information science
General
• DiVincenzo's criteria
• NISQ era
• Quantum computing
• timeline
• Quantum information
• Quantum programming
• Quantum simulation
• Qubit
• physical vs. logical
• Quantum processors
• cloud-based
Theorems
• Bell's
• Eastin–Knill
• Gleason's
• Gottesman–Knill
• Holevo's
• Margolus–Levitin
• No-broadcasting
• No-cloning
• No-communication
• No-deleting
• No-hiding
• No-teleportation
• PBR
• Threshold
• Solovay–Kitaev
• Purification
Quantum
communication
• Classical capacity
• entanglement-assisted
• quantum capacity
• Entanglement distillation
• Monogamy of entanglement
• LOCC
• Quantum channel
• quantum network
• Quantum teleportation
• quantum gate teleportation
• Superdense coding
Quantum cryptography
• Post-quantum cryptography
• Quantum coin flipping
• Quantum money
• Quantum key distribution
• BB84
• SARG04
• other protocols
• Quantum secret sharing
Quantum algorithms
• Amplitude amplification
• Bernstein–Vazirani
• Boson sampling
• Deutsch–Jozsa
• Grover's
• HHL
• Hidden subgroup
• Quantum annealing
• Quantum counting
• Quantum Fourier transform
• Quantum optimization
• Quantum phase estimation
• Shor's
• Simon's
• VQE
Quantum
complexity theory
• BQP
• EQP
• QIP
• QMA
• PostBQP
Quantum
processor benchmarks
• Quantum supremacy
• Quantum volume
• Randomized benchmarking
• XEB
• Relaxation times
• T1
• T2
Quantum
computing models
• Adiabatic quantum computation
• Continuous-variable quantum information
• One-way quantum computer
• cluster state
• Quantum circuit
• quantum logic gate
• Quantum machine learning
• quantum neural network
• Quantum Turing machine
• Topological quantum computer
Quantum
error correction
• Codes
• CSS
• quantum convolutional
• stabilizer
• Shor
• Bacon–Shor
• Steane
• Toric
• gnu
• Entanglement-assisted
Physical
implementations
Quantum optics
• Cavity QED
• Circuit QED
• Linear optical QC
• KLM protocol
Ultracold atoms
• Optical lattice
• Trapped-ion QC
Spin-based
• Kane QC
• Spin qubit QC
• NV center
• NMR QC
Superconducting
• Charge qubit
• Flux qubit
• Phase qubit
• Transmon
Quantum
programming
• OpenQASM-Qiskit-IBM QX
• Quil-Forest/Rigetti QCS
• Cirq
• Q#
• libquantum
• many others...
• Quantum information science
• Quantum mechanics topics
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
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Wikipedia
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Quantum KZ equations
In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the N-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter q approaches 1, the N-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics.
See also
• Quantum affine algebras
• Yang–Baxter equation
• Quantum group
• Affine Hecke algebra
• Kac–Moody algebra
• Two-dimensional conformal field theory
References
• Frenkel, I. B.; Reshetikhin, N. Yu. (1992), "Quantum affine algebras and holonomic difference equations", Comm. Math. Phys., 146 (1): 1–60, Bibcode:1992CMaPh.146....1F, doi:10.1007/BF02099206, S2CID 119818318
• Etingof, Pavel I.; Frenkel, Igor; Kirillov, Alexander A. (1998), Lectures on representation theory and Knizhnik–Zamolodchikov equations, Mathematical Surveys and Monographs, vol. 58, American Mathematical Society, ISBN 0821804960
• Jimbo, Michio; Miwa, Tetsuji (1995), Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, ISBN 0-8218-0320-4
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Wikipedia
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Quantum Markov chain
In mathematics, the quantum Markov chain is a reformulation of the ideas of a classical Markov chain, replacing the classical definitions of probability with quantum probability.
Introduction
Very roughly, the theory of a quantum Markov chain resembles that of a measure-many automaton, with some important substitutions: the initial state is to be replaced by a density matrix, and the projection operators are to be replaced by positive operator valued measures.
Formal statement
More precisely, a quantum Markov chain is a pair $(E,\rho )$ with $\rho $ a density matrix and $E$ a quantum channel such that
$E:{\mathcal {B}}\otimes {\mathcal {B}}\to {\mathcal {B}}$
is a completely positive trace-preserving map, and ${\mathcal {B}}$ a C*-algebra of bounded operators. The pair must obey the quantum Markov condition, that
$\operatorname {Tr} \rho (b_{1}\otimes b_{2})=\operatorname {Tr} \rho E(b_{1},b_{2})$
for all $b_{1},b_{2}\in {\mathcal {B}}$.
See also
• Quantum walk
References
• Gudder, Stanley. "Quantum Markov chains." Journal of Mathematical Physics 49.7 (2008): 072105.
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Wikipedia
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PCP theorem
In computational complexity theory, the PCP theorem (also known as the PCP characterization theorem) states that every decision problem in the NP complexity class has probabilistically checkable proofs (proofs that can be checked by a randomized algorithm) of constant query complexity and logarithmic randomness complexity (uses a logarithmic number of random bits).
Not to be confused with Post correspondence problem.
The PCP theorem says that for some universal constant K, for every n, any mathematical proof for a statement of length n can be rewritten as a different proof of length poly(n) that is formally verifiable with 99% accuracy by a randomized algorithm that inspects only K letters of that proof.
The PCP theorem is the cornerstone of the theory of computational hardness of approximation, which investigates the inherent difficulty in designing efficient approximation algorithms for various optimization problems. It has been described by Ingo Wegener as "the most important result in complexity theory since Cook's theorem"[1] and by Oded Goldreich as "a culmination of a sequence of impressive works […] rich in innovative ideas".[2]
Formal statement
The PCP theorem states that
NP = PCP[O(log n), O(1)],
where PCP[r(n), q(n)] is the class of problems for which a probabilistically checkable proof of a solution can be given, such that the proof can be checked in polynomial time using r(n) bits of randomness and by reading q(n) bits of the proof, correct proofs are always accepted, and incorrect proofs are rejected with probability at least 1/2. n is the length in bits of the description of a problem instance. Note further that the verification algorithm is non-adaptive: the choice of bits of the proof to check depend only on the random bits and the description of the problem instance, not the actual bits of the proof.
PCP and hardness of approximation
An alternative formulation of the PCP theorem states that the maximum fraction of satisfiable constraints of a constraint satisfaction problem is NP-hard to approximate within some constant factor.[3]
Formally, for some constants q and α < 1, the following promise problem (Lyes, Lno) is an NP-hard decision problem:
• Lyes = {Φ: all constraints in Φ are simultaneously satisfiable}
• Lno = {Φ: every assignment satisfies fewer than an α fraction of Φ's constraints},
where Φ is a constraint satisfaction problem (CSP) over a Boolean alphabet with at most q variables per constraint.
The connection to the class PCP mentioned above can be seen by noticing that checking a constant number of bits q in a proof can be seen as evaluating a constraint in q Boolean variables on those bits of the proof. Since the verification algorithm uses O(log n) bits of randomness, it can be represented as a CSP as described above with poly(n) constraints. The other characterisation of the PCP theorem then guarantees the promise condition with α = 1/2: if the NP problem's answer is yes, then every constraint (which corresponds to a particular value for the random bits) has a satisfying assignment (an acceptable proof); otherwise, any proof should be rejected with probability at least 1/2, which means any assignment must satisfy fewer than 1/2 of the constraints (which means it will be accepted with probability lower than 1/2). Therefore, an algorithm for the promise problem would be able to solve the underlying NP problem, and hence the promise problem must be NP hard.
As a consequence of this theorem, it can be shown that the solutions to many natural optimization problems including maximum boolean formula satisfiability, maximum independent set in graphs, and the shortest vector problem for lattices cannot be approximated efficiently unless P = NP. This can be done by reducing the problem of approximating a solution to such problems to a promise problem of the above form. These results are sometimes also called PCP theorems because they can be viewed as probabilistically checkable proofs for NP with some additional structure.
Proof
A proof of a weaker result, NP ⊆ PCP[n3, 1] is given in one of the lectures of Dexter Kozen.[4]
History
The PCP theorem is the culmination of a long line of work on interactive proofs and probabilistically checkable proofs. The first theorem relating standard proofs and probabilistically checkable proofs is the statement that NEXP ⊆ PCP[poly(n), poly(n)], proved by Babai, Fortnow & Lund (1990).
Origin of the initials
The notation PCPc(n), s(n)[r(n), q(n)] is explained at probabilistically checkable proof. The notation is that of a function that returns a certain complexity class. See the explanation mentioned above.
The name of this theorem (the "PCP theorem") probably comes either from "PCP" meaning "probabilistically checkable proof", or from the notation mentioned above (or both).
First theorem [in 1990]
Subsequently, the methods used in this work were extended by Babai, Lance Fortnow, Levin, and Szegedy in 1991 (Babai et al. 1991), Feige, Goldwasser, Lund, Safra, and Szegedy (1991), and Arora and Safra in 1992 (Arora & Safra 1992) to yield a proof of the PCP theorem by Arora, Lund, Motwani, Sudan, and Szegedy in 1998 (Arora et al. 1998).
The 2001 Gödel Prize was awarded to Sanjeev Arora, Uriel Feige, Shafi Goldwasser, Carsten Lund, László Lovász, Rajeev Motwani, Shmuel Safra, Madhu Sudan, and Mario Szegedy for work on the PCP theorem and its connection to hardness of approximation.
In 2005 Irit Dinur discovered a significantly simpler proof of the PCP theorem, using expander graphs.[5] She received the 2019 Gödel Prize for this. [6]
Quantum analog
In 2012, Thomas Vidick and Tsuyoshi Ito published a result[7] that showed a "strong limitation on the ability of entangled provers to collude in a multiplayer game". This could be a step toward proving the quantum analogue of the PCP theorem, since when the result[7] was reported in the media,[8][9] professor Dorit Aharonov called it "the quantum analogue of an earlier paper on multiprover interactive proofs" that "basically led to the PCP theorem".[9]
In 2018, Thomas Vidick and Anand Natarajan proved[10] a games variant of quantum PCP theorem under randomized reduction. It states that QMA ⊆ MIP*[log(n), 1, 1/2], where MIP*[f(n), c, s] is a complexity class of multi-prover quantum interactive proofs systems with f(n)-bit classical communications, and the completeness is c and the soundness is s. They also showed that the Hamiltonian version of a quantum PCP conjecture, namely a local Hamiltonian problem with constant promise gap c − s is QMA-hard, implies the games quantum PCP theorem.
NLTS conjecture is a fundamental unresolved obstacle and precursor to a quantum analog of PCP.[11]
Notes
1. Ingo Wegener (2005). Complexity Theory: Exploring the Limits of Efficient Algorithms. Springer. p. 161. ISBN 978-3-540-21045-0.
2. Oded Goldreich (2008). Computational Complexity: A Conceptual Perspective. Cambridge University Press. p. 405. ISBN 978-0-521-88473-0.
3. Arora, Sanjeev; Barak, Boaz (2009). Computational Complexity: a Modern Approach (PDF) (Draft). Cambridge University Press.
4. Kozen, Dexter C. (2006). Theory of Computation. Texts in Computer Science. London: Springer-Verlag. pp. 119–127. ISBN 9781846282973.
5. See the 2005 preprint, ECCC TR05-046. The authoritative version of the paper is Dinur (2007).
6. EATSC 2019 Gödel Prize, retrieved 2019-09-11.
7. Ito, Tsuyoshi; Vidick, Thomas (2012). "A multi-prover interactive proof for NEXP sound against entangled provers". 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20–23, 2012. IEEE Computer Society. pp. 243–252. arXiv:1207.0550. doi:10.1109/FOCS.2012.11.
8. Hardesty, Larry (2012-07-30). "MIT News Release: 10-year-old problem in theoretical computer science falls". MIT News Office. Archived from the original on 2014-02-02. Retrieved 2012-08-10. Interactive proofs are the basis of cryptographic systems now in wide use, but for computer scientists, they're just as important for the insight they provide into the complexity of computational problems.
9. Hardesty, Larry (2012-07-31). "10-year-old problem in theoretical computer science falls". MIT News Office. Archived from the original on 2012-08-01. Retrieved 2012-08-10. Dorit Aharonov, a professor of computer science and engineering at Hebrew University in Jerusalem, says that Vidick and Ito's paper is the quantum analogue of an earlier paper on multiprover interactive proofs that "basically led to the PCP theorem, and the PCP theorem is no doubt the most important result of complexity in the past 20 years." Similarly, she says, the new paper "could be an important step toward proving the quantum analogue of the PCP theorem, which is a major open question in quantum complexity theory."
10. Natarajan, A.; Vidick, T. (October 2018). "Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA". 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS). pp. 731–742. arXiv:1801.03821. Bibcode:2018arXiv180103821N. doi:10.1109/FOCS.2018.00075. ISBN 978-1-5386-4230-6. S2CID 53062680.
11. "On the NLTS Conjecture". Simons Institute for the Theory of Computing. 2021-06-30. Retrieved 2022-08-08.
References
• Arora, Sanjeev; Lund, Carsten; Motwani, Rajeev; Sudan, Madhu; Szegedy, Mario (1998), "Proof verification and the hardness of approximation problems", Journal of the ACM, 45 (3): 501–555, doi:10.1145/278298.278306, S2CID 8561542.
• Arora, Sanjeev; Safra, Shmuel (1992), "Approximating clique is NP-complete", In Proceedings of the 33rd IEEE Symposium on Foundations on Computer Science, 41 (1): 2–13
• Arora, Sanjeev; Safra, Shmuel (1998), "Probabilistic checking of proofs: A new characterization of NP", Journal of the ACM, 45 (1): 70–122, doi:10.1145/273865.273901, S2CID 751563.
• Babai, László; Fortnow, Lance; Levin, Leonid; Szegedy, Mario (1991), "Checking computations in polylogarithmic time", STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of computing, ACM, pp. 21–32, ISBN 978-0-89791-397-3.
• Babai, László; Fortnow, Lance; Lund, Carsten (1990), "Nondeterministic exponential time has two-prover interactive protocols", SFCS '90: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, IEEE Computer Society, pp. 16–25, ISBN 978-0-8186-2082-9.
• Dinur, Irit (2007), "The PCP theorem by gap amplification", Journal of the ACM, 54 (3): 12–es, doi:10.1145/1236457.1236459, S2CID 53244523.
• Feige, Uriel; Goldwasser, Shafi; Lovász, László; Safra, Shmuel; Szegedy, Mario (1996), "Interactive proofs and the hardness of approximating cliques" (PDF), Journal of the ACM, ACM, 43 (2): 268–292, doi:10.1145/226643.226652, ISSN 0004-5411.
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Wikipedia
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Quantum logic
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical mechanics forms a Boolean algebra, but the structure of experimental tests in quantum mechanics forms a much more complicated structure.
Part of a series of articles about
Quantum mechanics
$i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle $
Schrödinger equation
• Introduction
• Glossary
• History
Background
• Classical mechanics
• Old quantum theory
• Bra–ket notation
• Hamiltonian
• Interference
Fundamentals
• Complementarity
• Decoherence
• Entanglement
• Energy level
• Measurement
• Nonlocality
• Quantum number
• State
• Superposition
• Symmetry
• Tunnelling
• Uncertainty
• Wave function
• Collapse
Experiments
• Bell's inequality
• Davisson–Germer
• Double-slit
• Elitzur–Vaidman
• Franck–Hertz
• Leggett–Garg inequality
• Mach–Zehnder
• Popper
• Quantum eraser
• Delayed-choice
• Schrödinger's cat
• Stern–Gerlach
• Wheeler's delayed-choice
Formulations
• Overview
• Heisenberg
• Interaction
• Matrix
• Phase-space
• Schrödinger
• Sum-over-histories (path integral)
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
Advanced topics
• Relativistic quantum mechanics
• Quantum field theory
• Quantum information science
• Quantum computing
• Quantum chaos
• EPR paradox
• Density matrix
• Scattering theory
• Quantum statistical mechanics
• Quantum machine learning
Scientists
• Aharonov
• Bell
• Bethe
• Blackett
• Bloch
• Bohm
• Bohr
• Born
• Bose
• de Broglie
• Compton
• Dirac
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• von Neumann
• Weyl
• Wien
• Wigner
• Zeeman
• Zeilinger
A number of other logics have also been proposed to analyze quantum-mechanical phenomena, unfortunately also under the name of "quantum logic(s)." They are not the subject of this article. For discussion of the similarities and differences between quantum logic and some of these competitors, see § Relationship to other logics.
Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher Hilary Putnam, at least at one point in his career. This thesis was an important ingredient in Putnam's 1968 paper "Is Logic Empirical?" in which he analysed the epistemological status of the rules of propositional logic. Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a material conditional; a common alternative is the system of linear logic, of which quantum logic is a fragment.
Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an orthocomplemented lattice. Quantum-mechanical observables and states can be defined in terms of functions on or to the lattice, giving an alternate formalism for quantum computations.
Introduction
The most notable difference between quantum logic and classical logic is the failure of the propositional distributive law:[1]
p and (q or r) = (p and q) or (p and r),
where the symbols p, q and r are propositional variables.
To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let[Note 1]
p = "the particle has momentum in the interval [0, +1⁄6]"
q = "the particle is in the interval [−1, 1]"
r = "the particle is in the interval [1, 3]"
We might observe that:
p and (q or r) = true
in other words, that the state of the particle is a weighted superposition of momenta between 0 and +1/6 and positions between −1 and +3.
On the other hand, the propositions "p and q" and "p and r" each assert tighter restrictions on simultaneous values of position and momentum than are allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So there are no states that can support either proposition, and
(p and q) or (p and r) = false
History and modern criticism
In his classic 1932 treatise Mathematical Foundations of Quantum Mechanics, John von Neumann noted that projections on a Hilbert space can be viewed as propositions about physical observables; that is, as potential yes-or-no questions an observer might ask about the state of a physical system, questions that could be settled by some measurement.[2] Principles for manipulating these quantum propositions were then called quantum logic by von Neumann and Birkhoff in a 1936 paper.[3]
George Mackey, in his 1963 book (also called Mathematical Foundations of Quantum Mechanics), attempted to axiomatize quantum logic as the structure of an orthocomplemented lattice, and recognized that a physical observable could be defined in terms of quantum propositions. Although Mackey's presentation still assumed that the orthocomplemented lattice is the lattice of closed linear subspaces of a separable Hilbert space,[4] Constantin Piron, Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space.[5]
Inspired by Hans Reichenbach's recent defence of general relativity, the philosopher Hilary Putnam popularized Mackey's work in two papers in 1968 and 1975,[6] in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist David Finkelstein.[7] Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.[6][8] Later, Putnam retracted his views, albeit with much less fanfare,[6] but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.[9] Their work proved fruitless, and now lies in poor repute.[10]
Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true[11]) that quantum logic is a logic, in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.[12][13] In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.[14] Tim Maudlin writes that quantum "logic 'solves' the [measurement] problem by making the problem impossible to state."[15]
The horse of quantum logic has been so thrashed, whipped and pummeled, and is so thoroughly deceased that...the question is not whether the horse will rise again, it is: how in the world did this horse get here in the first place? The tale of quantum logic is not the tale of a promising idea gone bad, it is rather the tale of the unrelenting pursuit of a bad idea. ...Many, many philosophers and physicists have become convinced that a change of logic (and most dramatically, the rejection of classical logic) will somehow help in understanding quantum theory, or is somehow suggested or forced on us by quantum theory. But quantum logic, even through its many incarnations and variations, both in technical form and in interpretation, has never yielded the goods.
— Maudlin, Hilary Putnam, pp. 184-185
Quantum logic remains in limited use among logicians as an extremely pathological counterexample (Dalla Chiara and Giuntini: "Why quantum logics? Simply because 'quantum logics are there!'").[16] Although the central insight to quantum logic remains mathematical folklore as an intuition pump for categorification, discussions rarely mention quantum logic.[17]
Quantum logic's best chance at revival is through the recent development of quantum computing, which has engendered a proliferation of new logics for formal analysis of quantum protocols and algorithms (see also § Relationship to other logics).[18] The logic may also find application in (computational) linguistics.
Algebraic structure
Quantum logic can be axiomatized as the theory of propositions modulo the following identities:[19]
• a=¬¬a
• ∨ is commutative and associative.
• There is a maximal element ⊤, and ⊤=b∨¬b for any b.
• a∨¬(¬a∨b)=a.
("¬" is the traditional notation for "not", "∨" the notation for "or", and "∧" the notation for "and".)
Some authors restrict to orthomodular lattices, which additionally satisfy the orthomodular law:[20]
• If ⊤=¬(¬a∨¬b)∨¬(a∨b) then a=b.
("⊤" is the traditional notation for truth and ""⊥" the traditional notation for falsity.)
Alternative formulations include propositions derivable via a natural deduction,[16] sequent calculus[21][22] or tableaux system.[23] Despite the relatively developed proof theory, quantum logic is not known to be decidable.[19]
Quantum logic as the logic of observables
The remainder of this article assumes the reader is familiar with the spectral theory of self-adjoint operators on a Hilbert space. However, the main ideas can be understood in the finite-dimensional case.
The logic of classical mechanics
The Hamiltonian formulations of classical mechanics have three ingredients: states, observables and dynamics. In the simplest case of a single particle moving in R3, the state space is the position-momentum space R6. An observable is some real-valued function f on the state space. Examples of observables are position, momentum or energy of a particle. For classical systems, the value f(x), that is the value of f for some particular system state x, is obtained by a process of measurement of f.
The propositions concerning a classical system are generated from basic statements of the form
"Measurement of f yields a value in the interval [a, b] for some real numbers a, b."
through the conventional arithmetic operations and pointwise limits. It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the Boolean algebra of Borel subsets of the state space. They thus obey the laws of classical propositional logic (such as de Morgan's laws) with the set operations of union and intersection corresponding to the Boolean conjunctives and subset inclusion corresponding to material implication.
In fact, a stronger claim is true: they must obey the infinitary logic Lω1,ω.
We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished orthocomplementation operation: The lattice operations of meet and join are respectively set intersection and set union. The orthocomplementation operation is set complement. Moreover, this lattice is sequentially complete, in the sense that any sequence {Ei}i of elements of the lattice has a least upper bound, specifically the set-theoretic union:
$\operatorname {LUB} (\{E_{i}\})=\bigcup _{i=1}^{\infty }E_{i}{\text{.}}$
The propositional lattice of a quantum mechanical system
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f on R, the following extension of f to operators can be made:
$f(A)=\int _{\mathbb {R} }f(\lambda )\,d\operatorname {E} (\lambda ).$
In case f is the indicator function of an interval [a, b], the operator f(A) is a self-adjoint projection onto the subspace of generalized eigenvectors of A with eigenvalue in [a,b]. That subspace can be interpreted as the quantum analogue of the classical proposition
• Measurement of A yields a value in the interval [a, b].
This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's Axiom VII:
• The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of H; the negation of a proposition V is the orthogonal complement V⊥.
The space Q of quantum propositions is also sequentially complete: any pairwise disjoint sequence{Vi}i of elements of Q has a least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1⊥. The least upper bound of {Vi}i is the closed internal direct sum.
Standard semantics
The standard semantics of quantum logic is that quantum logic is the logic of projection operators in a separable Hilbert or pre-Hilbert space, where an observable p is associated with the set of quantum states for which p (when measured) has eigenvalue 1. From there,
• ¬p is the orthogonal complement of p (since for those states, the probability of observing p, P(p) = 0),
• p∧q is the intersection of p and q, and
• p∨q = ¬(¬p∧¬q) refers to states that superpose p and q.
This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the Solèr theorem.[24] Although much of the development of quantum logic has been motivated by the standard semantics, it is not the characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.[16]
Differences with classical logic
The structure of Q immediately points to a difference with the partial order structure of a classical proposition system. In the classical case, given a proposition p, the equations
⊤=p∨q and
⊥=p∧q
have exactly one solution, namely the set-theoretic complement of p. In the case of the lattice of projections there are infinitely many solutions to the above equations (any closed, algebraic complement of p solves it; it need not be the orthocomplement).
More generally, propositional valuation has unusual properties in quantum logic. An orthocomplemented lattice admitting a total lattice homomorphism to {⊥,⊤} must be Boolean. A standard workaround is to study maximal partial homomorphisms q with a filtering property:
if a≤b and q(a)=⊤, then q(b)=⊤.[10]
Failure of distributivity
Expressions in quantum logic describe observables using a syntax that resembles classical logic. However, unlike classical logic, the distributive law a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) fails when dealing with noncommuting observables, such as position and momentum. This occurs because measurement affects the system, and measurement of whether a disjunction holds does not measure which of the disjuncts is true.
For example, consider a simple one-dimensional particle with position denoted by x and momentum by p, and define observables:
• a — |p| ≤ 1 (in some units)
• b — x < 0
• c — x ≥ 0
Now, position and momentum are Fourier transforms of each other, and the Fourier transform of a square-integrable nonzero function with a compact support is entire and hence does not have non-isolated zeroes. Therefore, there is no wave function that is both normalizable in momentum space and vanishes on precisely x ≥ 0. Thus, a ∧ b and similarly a ∧ c are false, so (a ∧ b) ∨ (a ∧ c) is false. However, a ∧ (b ∨ c) equals a, which is certainly not false (there are states for which it is a viable measurement outcome). Moreover: if the relevant Hilbert space for the particle's dynamics only admits momenta no greater than 1, then a is true.
To understand more, let p1 and p2 be the momenta for the restriction of the particle wave function to x < 0 and x ≥ 0 respectively (with the wave function zero outside of the restriction). Let |p|↾>1 be the restriction of |p| to momenta that are (in absolute value) >1.
(a ∧ b) ∨ (a ∧ c) corresponds to states with |p1|↾>1 = |p2|↾>1 = 0 (this holds even if we defined p differently so as to make such states possible; also, a ∧ b corresponds to |p1|↾>1=0 and p2=0). As an operator, p=p1+p2, and nonzero |p1|↾>1 and |p2|↾>1 might interfere to produce zero |p|↾>1. Such interference is key to the richness of quantum logic and quantum mechanics.
Relationship to quantum measurement
Mackey observables
Given a orthocomplemented lattice Q, a Mackey observable φ is a countably additive homomorphism from the orthocomplemented lattice of Borel subsets of R to Q. In symbols, this means that for any sequence {Si}i of pairwise disjoint Borel subsets of R, {φ(Si)}i are pairwise orthogonal propositions (elements of Q) and
$\varphi \left(\bigcup _{i=1}^{\infty }S_{i}\right)=\sum _{i=1}^{\infty }\varphi (S_{i}).$
Equivalently, a Mackey observable is a projection-valued measure on R.
Theorem (Spectral theorem). If Q is the lattice of closed subspaces of Hilbert H, then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H.
Quantum probability measures
A quantum probability measure is a function P defined on Q with values in [0,1] such that P("⊥)=0, P(⊤)=1 and if {Ei}i is a sequence of pairwise orthogonal elements of Q then
$\operatorname {P} \!\left(\bigvee _{i=1}^{\infty }E_{i}\right)=\sum _{i=1}^{\infty }\operatorname {P} (E_{i}).$
Every quantum probability measure on the closed subspaces of a Hilbert space is induced by a density matrix — a nonnegative operator of trace 1. Formally,
Theorem.[25] Suppose Q is the lattice of closed subspaces of a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure P on Q there exists a unique trace class operator S such that
$\operatorname {P} (E)=\operatorname {Tr} (SE)$
for any self-adjoint projection E in Q.
Relationship to other logics
Quantum logic embeds into linear logic[26] and the modal logic B.[16] Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and attempt to graft desirable features of an extension of classical logic thereonto; the results then necessarily embed quantum logic.[27][28]
The orthocomplemented lattice of any set of quantum propositions can be embedded into a Boolean algebra, which is then amenable to classical logic.[29]
Limitations
Although many treatments of quantum logic assume that the underlying lattice must be orthomodular, such logics cannot handle multiple interacting quantum systems. In an example due to Foulis and Randall, there are orthomodular propositions with finite-dimensional Hilbert models whose pairing admits no orthomodular model.[8] Likewise, quantum logic with the orthomodular law fails the deduction theorem.[30][31]
Quantum logic admits no reasonable material conditional; any connective that is monotone in a certain technical sense reduces the class of propositions to a Boolean algebra.[32] Consequently, quantum logic struggles to represent the passage of time.[26] One possible workaround is the theory of quantum filtrations developed in the late 1970s and 1980s by Belavkin.[33][34] It is known, however, that System BV, a deep inference fragment of linear logic that is very close to quantum logic, can handle arbitrary discrete spacetimes.[35]
See also
• Fuzzy logic
• HPO formalism (An approach to temporal quantum logic)
• Linear logic
• Mathematical formulation of quantum mechanics
• Multi-valued logic
• Quantum Bayesianism
• Quantum cognition
• Quantum contextuality
• Quantum field theory
• Quantum probability
• Quasi-set theory
• Solèr's theorem
• Vector logic
Notes
1. Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that are feasible. See § Quantum logic as the logic of observables et seq. for details.
Citations
1. Peter Forrest, "Quantum logic" in Routledge Encyclopedia of Philosophy, vol. 7, 1998. p. 882ff: "[Quantum logic] differs from the standard sentential calculus....The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity."
2. von Neumann 1932.
3. Birkhoff & von Neumann 1936.
4. Mackey 1963.
5. Piron:
• C. Piron, "Axiomatique quantique" (in French), Helvetica Physica Acta vol. 37, 1964. DOI: 10.5169/seals-113494.
• Piron 1976.
Ludwig:
• Günther Ludwig, "Attempt of an Axiomatic Foundation of Quantum Mechanics and More General Theories" II, Commun. Math. Phys., vol. 4, 1967. pp. 331-348.
• Ludwig 1954
6. Maudlin 2005.
7. Putnam 1969.
8. Wilce.
9. T. A. Brody, "On Quantum Logic", Foundations of Physics, vol. 14, no. 5, 1984. pp. 409-430.
10. Bacciagaluppi 2009.
11. Dalla Chiara & Giuntini 2002, p. 94: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."
12. Maudlin 2005, p. 159-161.
13. Brody 1984.
14. Brody 1984, pp. 428–429.
15. Maudlin 2005, p. 174.
16. Dalla Chiara & Giuntini 2002.
17. Terry Tao, "Venn and Euler type diagrams for vector spaces and abelian groups" on What's New (blog), 2021.
18. Dalla Chiara, Giuntini & Leporini 2003.
19. Megill 2019.
20. Kalmbach 1974 and Kalmbach 1983
21. N.J. Cutland; P.F. Gibbins (Sep 1982). "A regular sequent calculus for Quantum Logic in which ∨ and ∧ are dual". Logique et Analyse. Nouvelle Série. 25 (99): 221–248. JSTOR 44084050.
• Hirokazu Nishimura (Jan 1994). "Proof theory for minimal quantum logic I". International Journal of Theoretical Physics. 33 (1): 103–113. Bibcode:1994IJTP...33..103N. doi:10.1007/BF00671616. S2CID 123183879.
• Hirokazu Nishimura (Jul 1994). "Proof theory for minimal quantum logic II". International Journal of Theoretical Physics. 33 (7): 1427–1443. Bibcode:1994IJTP...33.1427N. doi:10.1007/bf00670687. S2CID 189850106.
22. Uwe Egly; Hans Tompits (1999). Gentzen-like Methods in Quantum Logic (PDF). 8th Int. Conf. on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX). SUNY Albany. CiteSeerX 10.1.1.88.9045.
23. Dalla Chiara & Giuntini 2002 and de Ronde, Domenech & Freytes. Despite suggestions otherwise in Josef Jauch, Foundations of Quantum Mechanics, Addison-Wesley Series in Advanced Physics; Addison-Wesley, 1968, this property cannot be used to deduce a vector space structure, because it is not peculiar to (pre-)Hilbert spaces. An analogous claim holds in most categories; see John Harding, "Decompositions in Quantum Logic," Transactions of the AMS, vol. 348, no. 5, 1996. pp. 1839-1862.
24. A. Gleason, "Measures on the Closed Subspaces of a Hilbert Space", Indiana University Mathematics Journal, vol. 6, no. 4, 1957. pp. 885-893. DOI: 10.1512/iumj.1957.6.56050. Reprinted in The Logico-Algebraic Approach to Quantum Mechanics, University of Western Ontario Series in Philosophy of Science 5a, ed. C. A. Hooker; D. Riedel, c. 1975-1979. pp. 123-133.
25. Vaughan Pratt, "Linear logic for generalized quantum mechanics," in Workshop on Physics and Computation (PhysComp '92) proceedings. See also the discussion at nLab, Revision 42, which cites G.D. Crown, "On some orthomodular posets of vector bundles," Journ. of Natural Sci. and Math., vol. 15 issue 1-2: pp. 11–25, 1975.
26. Baltag & Smets 2006.
27. Baltag et al. 2014.
28. Jeffery Bub and William Demopoulos, "The Interpretation of Quantum Mechanics," in Logical and Epistemological Studies in Contemporary Physics, Boston Studies in the Philosophy of Science 13, ed. Robert S. Cohen and Marx W. Wartofsky; D. Riedel, 1974. pp. 92-122. DOI: 10.1007/978-94-010-2656-7. ISBN 978-94-010-2656-7.
29. Kalmbach 1981.
30. Kalmbach, G. (1981). "Omologic as a Hilbert Type Calculus". In Beltrametti, E. (ed.). Current Issues in Quantum Logic. Plenum Press. pp. 333–340.
31. Román, L.; Rumbos, B. (1991). "Quantum logic revisited" (PDF). Foundations of Physics. 21 (6): 727–734. Bibcode:1991FoPh...21..727R. doi:10.1007/BF00733278. S2CID 123383431.
• V. P. Belavkin (1978). "Optimal quantum filtration of Makovian signals". Problems of Control and Information Theory (in Russian). 7 (5): 345–360.
• V. P. Belavkin (1992). "Quantum stochastic calculus and quantum nonlinear filtering". Journal of Multivariate Analysis. 42 (2): 171–201. arXiv:math/0512362. doi:10.1016/0047-259X(92)90042-E. S2CID 3909067.
32. Luc Bouten; Ramon van Handel; Matthew R. James (2009). "A discrete invitation to quantum filtering and feedback control". SIAM Review. 51 (2): 239–316. arXiv:math/0606118. Bibcode:2009SIAMR..51..239B. doi:10.1137/060671504. S2CID 10435983.
33. Richard Blute, Alessio Guglielmi, Ivan T. Ivanov, Prakash Panangaden, Lutz Straßburger, "A Logical Basis for Quantum Evolution and Entanglement" in Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday; Springer, 2014. pp. 90-107. DOI: 10.1007/978-3-642-54789-8_6. HAL 01092279.
Sources
Historical works
Organized chronologically
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, trans. Robert T. Beyer, ed. Nicholas A. Wheeler; Princeton University Press, 2018 (original 1932). pp. 160-164. JSTOR j.ctt1wq8zhp. 1955 edition available at the Internet Archive.
• G. Birkhoff and J. von Neumann, "The Logic of Quantum Mechanics," Annals of Mathematics, series II, vol. 37, issue 4, pp. 823–843, 1936. JSTOR 1968621. DOI 10.2307/1968621.
• G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963. HathiTrust 2027/mdp.39015001329567.
• H. Putnam, Is Logic Empirical?, Boston Studies in the Philosophy of Science V, ed. Robert S. Cohen and Marx W. Wartofsky, 1969.
• G. Kalmbach Orthomodular Logic, Z. Logik und Grundl. Math., vol. 20, 1974, pp. 395-406.
• G. Kalmbach Orthomodular Logic as a Hilbert Type Calculus, in Current Issues in Quantum Logic, Plenum Press, New York, ed. E. Beltrametti et al., 1981, pp. 333-340
• G. Kalmbach Orthomodular Lattices, Academic Press, London, 1983
Modern philosophical perspectives
• Guido Bacciagaluppi, "Is Logic Empirical?", in Handbook of Quantum Logic and Quantum Structures: Quantum Logic, ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.
• Tim Maudlin, "The Tale of Quantum Logic" in Hilary Putnam; Cambridge University Press "Contemporary Philosophy in Focus" series, 2005. DOI: 10.1017/CBO9780511614187.006 ISBN 9780521012546.
• de Ronde, C.; Domenech, G.; Freytes, H. "Quantum Logic in Historical and Philosophical Perspective". Internet Encyclopedia of Philosophy.
• Wilce, Alexander. "Quantum Logic and Probability Theory". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
Mathematical study and computational applications
• A. Baltag and S. Smets, "LQP: The Dynamic Logic of Quantum Information", Mathematical Structures in Computer Science, vol. 16, issue 3, pp. 491-525, 2006. DOI 10.1017/S0960129506005299 arXiv 2110.01361
• A. Baltag, J. Bergfeld, K. Kishida, J. Sack, S. Smets and S. Zhong, "PLQP & Company: Decidable Logics for Quantum Algorithms", International Journal of Theoretical Physics, vol. 53, issue 10, pp. 3628-3647, 2014.
• M. L. Dalla Chiara and R. Giuntini, "Quantum Logics", in Handbook of Philosophical Logic, vol. 6, D. Gabbay and F. Guenthner (eds.), Kluwer, 2002. arXiv quant-ph/0101028
• M. L. Dalla Chiara, R. Giuntini, and R. Leporini, "Quantum Computational Logics: A Survey", in Trends in Logic, vol. 21, V. F. Hendricks and J. Malinowski (eds.), Springer, 2003. arXiv quant-ph/0305029
• Norman Megill, Quantum Logic Explorer at Metamath, 2019.
• N. Papanikolaou, "Reasoning Formally About Quantum Systems: An Overview", ACM SIGACT News, 36(3), 2005. pp. 51–66. arXiv cs/0508005.
Quantum foundations
• D. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.
• Günther Ludwig, Der Grundlagen der Quantenmechanik (in German), Springer, 1954. The definitive work. Released in English as:
• Günther Ludwig, Foundations of Quantum Mechanics, vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.
• Günther Ludwig, An Axiomatic Basis for Quantum Mechanics, vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI: 10.1007/978-3-642-70029-3. ISBN 978-3-642-70029-3.
• Quantum Logic at the nLab
• C. Piron, Foundations of Quantum Physics, W. A. Benjamin, 1976.
Look up quantum logic in Wiktionary, the free dictionary.
Quantum mechanics
Background
• Introduction
• History
• Timeline
• Classical mechanics
• Old quantum theory
• Glossary
Fundamentals
• Born rule
• Bra–ket notation
• Complementarity
• Density matrix
• Energy level
• Ground state
• Excited state
• Degenerate levels
• Zero-point energy
• Entanglement
• Hamiltonian
• Interference
• Decoherence
• Measurement
• Nonlocality
• Quantum state
• Superposition
• Tunnelling
• Scattering theory
• Symmetry in quantum mechanics
• Uncertainty
• Wave function
• Collapse
• Wave–particle duality
Formulations
• Formulations
• Heisenberg
• Interaction
• Matrix mechanics
• Schrödinger
• Path integral formulation
• Phase space
Equations
• Dirac
• Klein–Gordon
• Pauli
• Rydberg
• Schrödinger
Interpretations
• Bayesian
• Consistent histories
• Copenhagen
• de Broglie–Bohm
• Ensemble
• Hidden-variable
• Local
• Many-worlds
• Objective collapse
• Quantum logic
• Relational
• Transactional
• Von Neumann-Wigner
Experiments
• Bell's inequality
• Davisson–Germer
• Delayed-choice quantum eraser
• Double-slit
• Franck–Hertz
• Mach–Zehnder interferometer
• Elitzur–Vaidman
• Popper
• Quantum eraser
• Stern–Gerlach
• Wheeler's delayed choice
Science
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Technology
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Wikipedia
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Quantum Trajectory Theory
Quantum Trajectory Theory (QTT) is a formulation of quantum mechanics used for simulating open quantum systems, quantum dissipation and single quantum systems.[1] It was developed by Howard Carmichael in the early 1990s around the same time as the similar formulation, known as the quantum jump method or Monte Carlo wave function (MCWF) method, developed by Dalibard, Castin and Mølmer.[2] Other contemporaneous works on wave-function-based Monte Carlo approaches to open quantum systems include those of Dum, Zoller and Ritsch, and Hegerfeldt and Wilser.[3]
QTT is compatible with the standard formulation of quantum theory, as described by the Schrödinger equation, but it offers a more detailed view.[4][1] The Schrödinger equation can be used to compute the probability of finding a quantum system in each of its possible states should a measurement be made. This approach is fundamentally statistical and is useful for predicting average measurements of large ensembles of quantum objects but it does not describe or provide insight into the behaviour of individual particles. QTT fills this gap by offering a way to describe the trajectories of individual quantum particles that obey the probabilities computed from the Schrödinger equation.[4][5] Like the quantum jump method, QTT applies to open quantum systems that interact with their environment.[1] QTT has become particularly popular since the technology has been developed to efficiently control and monitor individual quantum systems as it can predict how individual quantum objects such as particles will behave when they are observed.[4]
Method
In QTT open quantum systems are modelled as scattering processes, with classical external fields corresponding to the inputs and classical stochastic processes corresponding to the outputs (the fields after the measurement process).[6] The mapping from inputs to outputs is provided by a quantum stochastic process that is set up to account for a particular measurement strategy (eg., photon counting, homodyne/heterodyne detection, etc).[7] The calculated system state as a function of time is known as a quantum trajectory, and the desired density matrix as a function of time may be calculated by averaging over many simulated trajectories.
Like other Monte Carlo approaches, QTT provides an advantage over direct master-equation approaches by reducing the number of computations required. For a Hilbert space of dimension N, the traditional master equation approach would require calculation of the evolution of N2 atomic density matrix elements, whereas QTT only requires N calculations. This makes it useful for simulating large open quantum systems.[8]
The idea of monitoring outputs and building measurement records is fundamental to QTT. This focus on measurement distinguishes it from the quantum jump method which has no direct connection to monitoring output fields. When applied to direct photon detection the two theories produce equivalent results. Where the quantum jump method predicts the quantum jumps of the system as photons are emitted, QTT predicts the "clicks" of the detector as photons are measured. The only difference is the viewpoint.[8]
QTT is also broader in its application than the quantum jump method as it can be applied to many different monitoring strategies including direct photon detection and heterodyne detection. Each different monitoring strategy offers a different picture of the system dynamics.[8]
Applications
There have been two distinct phases of applications for QTT. Like the quantum jump method, QTT was first used for computer simulations of large quantum systems. These applications exploit its ability to significantly reduce the size of computations, which was especially necessary in the 1990s when computing power was very limited.[2][9][10]
The second phase of application has been catalysed by the development of technologies to precisely control and monitor single quantum systems. In this context QTT is being used to predict and guide single quantum system experiments including those contributing to the development of quantum computers.[1][11][12][13][14][15][5]
Quantum measurement problem
QTT addresses one aspect of the measurement problem in quantum mechanics by providing a detailed description of the intermediate steps through which a quantum state approaches the final, measured state during the so-called "collapse of the wave function". It reconciles the concept of a quantum jump with the smooth evolution described by the Schrödinger equation. The theory suggests that "quantum jumps" are not instantaneous but happen in a coherently driven system as a smooth transition through a series of superposition states.[5] This prediction was tested experimentally in 2019 by a team at Yale University led by Michel Devoret and Zlatko Minev, in collaboration with Carmichael and others at Yale University and the University of Auckland. In their experiment they used a superconducting artificial atom to observe a quantum jump in detail, confirming that the transition is a continuous process that unfolds over time. They were also able to detect when a quantum jump was about to occur and intervene to reverse it, sending the system back to the state in which it started.[11] This experiment, inspired and guided by QTT, represents a new level of control over quantum systems and has potential applications in correcting errors in quantum computing in the future.[11][16][17][18][5][1]
References
1. Ball, Phillip (28 March 2020). "Reality in the making". New Scientist: 35–38.
2. Mølmer, K.; Castin, Y.; Dalibard, J. (1993). "Monte Carlo wave-function method in quantum optics". Journal of the Optical Society of America B. 10 (3): 524. Bibcode:1993JOSAB..10..524M. doi:10.1364/JOSAB.10.000524. S2CID 85457742.
3. The associated primary sources are, respectively:
• Dalibard, Jean; Castin, Yvan; Mølmer, Klaus (February 1992). "Wave-function approach to dissipative processes in quantum optics". Physical Review Letters. 68 (5): 580–583. arXiv:0805.4002. Bibcode:1992PhRvL..68..580D. doi:10.1103/PhysRevLett.68.580. PMID 10045937.
• Carmichael, Howard (1993). An Open Systems Approach to Quantum Optics. Springer-Verlag. ISBN 978-0-387-56634-4.
• Dum, R.; Zoller, P.; Ritsch, H. (1992). "Monte Carlo simulation of the atomic master equation for spontaneous emission". Physical Review A. 45 (7): 4879–4887. Bibcode:1992PhRvA..45.4879D. doi:10.1103/PhysRevA.45.4879. PMID 9907570.
• Hegerfeldt, G. C.; Wilser, T. S. (1992). "Ensemble or Individual System, Collapse or no Collapse: A Description of a Single Radiating Atom". In H.D. Doebner; W. Scherer; F. Schroeck, Jr. (eds.). Classical and Quantum Systems (PDF). Proceedings of the Second International Wigner Symposium. World Scientific. pp. 104–105.
4. Ball, Philip. "The Quantum Theory That Peels Away the Mystery of Measurement". Quanta Magazine. Retrieved 2020-08-14.
5. "Collaborating with the world's best to answer century-old mystery in quantum theory" (PDF). 2019 Dodd-Walls Centre Annual Report: 20–21.
6. "Howard Carmichael – Physik-Schule". physik.cosmos-indirekt.de (in German). Retrieved 2020-08-14.
7. "Dr Howard Carmichael - The University of Auckland". unidirectory.auckland.ac.nz. Retrieved 2020-08-14.
8. "Quantum optics. Proceedings of the XXth Solvay conference on physics, Brussels, November 6–9, 1991". Physics Reports. 1991.
9. L. Horvath and H. J. Carmichael (2007). "Effect of atomic beam alignment on photon correlation measurements in cavity QED". Physical Review A. 76, 043821 (4): 043821. arXiv:0704.1686. Bibcode:2007PhRvA..76d3821H. doi:10.1103/PhysRevA.76.043821. S2CID 56107461.
10. R. Chrétien (2014) "Laser cooling of atoms: Monte-Carlo wavefunction simulations" Masters Thesis.
11. Ball, Philip. "Quantum Leaps, Long Assumed to Be Instantaneous, Take Time". Quanta Magazine. Retrieved 2020-08-27.
12. Wiseman, H. (2011). Quantum Measurement and Control. Cambridge University Press.
13. K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi (2014). "Observing single quantum trajectories of a superconducting quantum bit". Nature. 502 (7470): 211–214. arXiv:1305.7270. doi:10.1038/nature12539. PMID 24108052. S2CID 3648689.{{cite journal}}: CS1 maint: multiple names: authors list (link)
14. N. Roch, M. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. Eddins, A. Korotkov, K. Whaley, M. Sarovar, and I. Siddiqi (2014). "Observation of measurement-induced entanglement and quantum trajectories of remote superconducting qubits". Physical Review Letters. 112, 170501-1-4, 2014. (17): 170501. arXiv:1402.1868. Bibcode:2014PhRvL.112q0501R. doi:10.1103/PhysRevLett.112.170501. PMID 24836225. S2CID 14481406 – via American Physical Society.{{cite journal}}: CS1 maint: multiple names: authors list (link)
15. P. Campagne-Ibarcq, P. Six, L. Bretheau, A. Sarlette, M. Mirrahimi, P. Rouchon, and B. Huard (2016). "Observing quantum state diffusion by heterodyne detection of fluorescence". Physical Review X. 6 (1): 011002. arXiv:1511.01415. Bibcode:2016PhRvX...6a1002C. doi:10.1103/PhysRevX.6.011002. S2CID 53548243.{{cite journal}}: CS1 maint: multiple names: authors list (link)
16. Shelton, Jim (3 June 2019). "Physicists can predict the jumps of Schrödinger's cat (and finally save it)". ScienceDaily. Retrieved 2020-08-25.
17. Dumé, Isabelle (7 June 2019). "To catch a quantum jump". Physics World. Retrieved 2020-08-25.
18. Lea, Robert (2019-06-03). "Predicting the leaps of Schrödinger's Cat". Medium. Retrieved 2020-08-25.
External links
• mcsolve Quantum jump (Monte Carlo) solver from QuTiP for Python.
• QuantumOptics.jl the quantum optics toolbox in Julia.
• Quantum Optics Toolbox for Matlab
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Wikipedia
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Quantum affine algebra
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized.
See also
• Quantum enveloping algebra
• Quantum KZ equations
• Littelmann path model
• Yangian
References
• Drinfeld, V. G. (1985), "Hopf algebras and the quantum Yang–Baxter equation", Doklady Akademii Nauk SSSR, 283 (5): 1060–1064, ISSN 0002-3264, MR 0802128
• Drinfeld, V. G. (1987), "A new realization of Yangians and of quantum affine algebras", Doklady Akademii Nauk SSSR, 296 (1): 13–17, ISSN 0002-3264, MR 0914215
• Frenkel, Igor B.; Reshetikhin, N. Yu. (1992), "Quantum affine algebras and holonomic difference equations", Communications in Mathematical Physics, 146 (1): 1–60, Bibcode:1992CMaPh.146....1F, doi:10.1007/BF02099206, ISSN 0010-3616, MR 1163666, S2CID 119818318
• Jimbo, Michio (1985), "A q-difference analogue of U(g) and the Yang-Baxter equation", Letters in Mathematical Physics, 10 (1): 63–69, Bibcode:1985LMaPh..10...63J, doi:10.1007/BF00704588, ISSN 0377-9017, MR 0797001, S2CID 123313856
• Jimbo, Michio; Miwa, Tetsuji (1995), Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC, ISBN 978-0-8218-0320-2, MR 1308712
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Wikipedia
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Quantum algebra
Quantum algebra is one of the top-level mathematics categories used by the arXiv. It is the study of noncommutative analogues and generalizations of commutative algebras, especially those arising in Lie theory.[1]
Subjects include:
• Quantum groups
• Skein theories
• Operadic algebra
• Diagrammatic algebra
• Quantum field theory
• Racks and quandles
See also
• Coherent states in mathematical physics
• Glossary of areas of mathematics
• Mathematics Subject Classification
• Ordered type system, a substructural type system
• Outline of mathematics
• Quantum logic
References
1. "What is quantum algebra?". mathoverflow.net. Retrieved 2018-01-22.
External links
• Quantum algebra at arxiv.org
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Wikipedia
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Quantum annealing
Quantum annealing (QA) is an optimization process for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations. Quantum annealing is used mainly for problems where the search space is discrete (combinatorial optimization problems) with many local minima; such as finding[1] the ground state of a spin glass or the traveling salesman problem. The term "quantum annealing" was first proposed in 1988 by B. Apolloni, N. Cesa Bianchi and D. De Falco as a quantum-inspired classical algorithm.[2][3] It was formulated in its present form by T. Kadowaki and H. Nishimori (ja) in "Quantum annealing in the transverse Ising model"[4] though an imaginary-time variant without quantum coherence had been discussed by A. B. Finnila, M. A. Gomez, C. Sebenik and J. D. Doll, in "Quantum annealing is a new method for minimizing multidimensional functions".[5]
Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states or essentially tunneling through peaks. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian (also see adiabatic quantum computation).[6] If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., diabatic quantum computation.[7][8] The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem. An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal.[9] Quantum annealing has also been proven to provide a fast Grover oracle for the square-root speedup in solving many NP-complete problems.[10]
Comparison to Simulated Annealing
Quantum annealing can be compared to simulated annealing, whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state. In quantum annealing, the strength of transverse field determines the quantum-mechanical probability to change the amplitudes of all states in parallel. Analytical[11] and numerical[12] evidence suggests that quantum annealing outperforms simulated annealing under certain conditions (see[13] for a careful analysis, and,[14] for a fully solvable model of quantum annealing to arbitrary target Hamiltonian and comparison of different computation approaches).
Quantum mechanics: analogy and advantage
The tunneling field is basically a kinetic energy term that does not commute with the classical potential energy part of the original glass. The whole process can be simulated in a computer using quantum Monte Carlo (or other stochastic technique), and thus obtain a heuristic algorithm for finding the ground state of the classical glass.
In the case of annealing a purely mathematical objective function, one may consider the variables in the problem to be classical degrees of freedom, and the cost functions to be the potential energy function (classical Hamiltonian). Then a suitable term consisting of non-commuting variable(s) (i.e. variables that have non-zero commutator with the variables of the original mathematical problem) has to be introduced artificially in the Hamiltonian to play the role of the tunneling field (kinetic part). Then one may carry out the simulation with the quantum Hamiltonian thus constructed (the original function + non-commuting part) just as described above. Here, there is a choice in selecting the non-commuting term and the efficiency of annealing may depend on that.
It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima.[15] Since thermal transition probabilities (proportional to $e^{-{\frac {\Delta }{k_{B}T}}}$, with $T$ the temperature and $k_{B}$ the Boltzmann constant) depend only on the height $\Delta $ of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier in 1989 by Ray, Chakrabarti & Chakrabarti,[1] the quantum tunneling probability through the same barrier (considered in isolation) depends not only on the height $\Delta $ of the barrier, but also on its width $w$ and is approximately given by $e^{-{\frac {{\sqrt {\Delta }}w}{\Gamma }}}$, where $\Gamma $ is the tunneling field.[16] This additional handle through the width $w$, in presence of quantum tunneling, can be of major help: If the barriers are thin enough (i.e. $w\ll {\sqrt {\Delta }}$), quantum fluctuations can surely bring the system out of the shallow local minima. For an $N$-spin glass, the barrier height $\Delta $ becomes of order $N$. For constant value of $w$ one gets $\tau $ proportional to $e^{\sqrt {N}}$ for the annealing time (instead of $\tau $ proportional to $e^{N}$ for thermal annealing), while $\tau $ can even become $N$-independent for cases where $w$ decreases as $1/{\sqrt {N}}$.[17][18]
It is speculated that in a quantum computer, such simulations would be much more efficient and exact than that done in a classical computer, because it can perform the tunneling directly, rather than needing to add it by hand. Moreover, it may be able to do this without the tight error controls needed to harness the quantum entanglement used in more traditional quantum algorithms. Some confirmation of this is found in exactly solvable models.[19][20]
Timeline of ideas related to quantum annealing in Ising spin glasses:
• 1989 Idea was presented that quantum fluctuations could help explore rugged energy landscapes of the classical Ising spin glasses by escaping from local minima (having tall but thin barriers) using tunneling;[1]
• 1998 Formulation of quantum annealing and numerical test demonstrating its advantages in Ising glass systems;[4]
• 1999 First experimental demonstration of quantum annealing in LiHoYF Ising glass magnets;[21]
• 2011 Superconducting-circuit quantum annealing machine built and marketed by D-Wave Systems.[22]
D-Wave implementations
In 2011, D-Wave Systems announced the first commercial quantum annealer on the market by the name D-Wave One and published a paper in Nature on its performance.[22] The company claims this system uses a 128 qubit processor chipset.[23] On May 25, 2011, D-Wave announced that Lockheed Martin Corporation entered into an agreement to purchase a D-Wave One system.[24] On October 28, 2011 USC's Information Sciences Institute took delivery of Lockheed's D-Wave One.
In May 2013 it was announced that a consortium of Google, NASA Ames and the non-profit Universities Space Research Association purchased an adiabatic quantum computer from D-Wave Systems with 512 qubits.[25][26] An extensive study of its performance as quantum annealer, compared to some classical annealing algorithms, is already available.[27]
In June 2014, D-Wave announced a new quantum applications ecosystem with computational finance firm 1QB Information Technologies (1QBit) and cancer research group DNA-SEQ to focus on solving real-world problems with quantum hardware.[28] As the first company dedicated to producing software applications for commercially available quantum computers, 1QBit's research and development arm has focused on D-Wave's quantum annealing processors and has successfully demonstrated that these processors are suitable for solving real-world applications.[29]
With demonstrations of entanglement published,[30] the question of whether or not the D-Wave machine can demonstrate quantum speedup over all classical computers remains unanswered. A study published in Science in June 2014, described as "likely the most thorough and precise study that has been done on the performance of the D-Wave machine"[31] and "the fairest comparison yet", attempted to define and measure quantum speedup. Several definitions were put forward as some may be unverifiable by empirical tests, while others, though falsified, would nonetheless allow for the existence of performance advantages. The study found that the D-Wave chip "produced no quantum speedup" and did not rule out the possibility in future tests.[32] The researchers, led by Matthias Troyer at the Swiss Federal Institute of Technology, found "no quantum speedup" across the entire range of their tests, and only inconclusive results when looking at subsets of the tests. Their work illustrated "the subtle nature of the quantum speedup question". Further work[33] has advanced understanding of these test metrics and their reliance on equilibrated systems, thereby missing any signatures of advantage due to quantum dynamics.
There are many open questions regarding quantum speedup. The ETH reference in the previous section is just for one class of benchmark problems. Potentially there may be other classes of problems where quantum speedup might occur. Researchers at Google, LANL, USC, Texas A&M, and D-Wave are working to find such problem classes.[34]
In December 2015, Google announced that the D-Wave 2X outperforms both simulated annealing and Quantum Monte Carlo by up to a factor of 100,000,000 on a set of hard optimization problems.[35]
D-Wave's architecture differs from traditional quantum computers. It is not known to be polynomially equivalent to a universal quantum computer and, in particular, cannot execute Shor's algorithm because Shor's algorithm is not a hillclimbing process. Shor's algorithm requires a universal quantum computer. During the Qubits 2021 conference held by D-Wave, it was announced[36] that the company is hard at work developing their first universal quantum computers, capable of running Shor's algorithm in addition to other gate-model algorithms such as QAOA and VQE.
"A cross-disciplinary introduction to quantum annealing-based algorithms" [37] presents an introduction to combinatorial optimization (NP-hard) problems, the general structure of quantum annealing-based algorithms and two examples of this kind of algorithms for solving instances of the max-SAT and Minimum Multicut problems, together with an overview of the quantum annealing systems manufactured by D-Wave Systems. Hybrid quantum-classic algorithms for large-scale discrete-continuous optimization problems were reported to illustrate the quantum advantage.[38][39]
References
1. Ray, P.; Chakrabarti, B. K.; Chakrabarti, A. (1989). "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations". Physical Review B. 39 (16): 11828–11832. Bibcode:1989PhRvB..3911828R. doi:10.1103/PhysRevB.39.11828. PMID 9948016.
2. Apolloni, Bruno; Cesa-Bianchi, Nicolo; De Falco, Diego (July 1988). "A numerical implementation of quantum annealing". Stochastic Processes, Physics and Geometry, Proceedings of the Ascona-Locarno Conference.
3. Apolloni, Bruno; Carvalho, Maria C.; De Falco, Diego (1989). "Quantum stochastic optimization". Stoc. Proc. Appl. 33 (2): 233–244. doi:10.1016/0304-4149(89)90040-9.
4. Kadowaki, T.; Nishimori, H. (1998). "Quantum annealing in the transverse Ising model". Phys. Rev. E. 58 (5): 5355. arXiv:cond-mat/9804280. Bibcode:1998PhRvE..58.5355K. doi:10.1103/PhysRevE.58.5355. S2CID 36114913. Archived from the original on 2013-08-11.
5. Finnila, A.B.; Gomez, M.A.; Sebenik, C.; Stenson, C.; Doll, J.D. (1994). "Quantum annealing: A new method for minimizing multidimensional functions". Chemical Physics Letters. 219 (5–6): 343–348. arXiv:chem-ph/9404003. Bibcode:1994CPL...219..343F. doi:10.1016/0009-2614(94)00117-0. S2CID 97302385.
6. Farhi, E.; Goldstone, J.; Gutmann, S.; Lapan, J.; Ludgren, A.; Preda, D. (2001). "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem". Science. 292 (5516): 472–5. arXiv:quant-ph/0104129. Bibcode:2001Sci...292..472F. doi:10.1126/science.1057726. PMID 11313487. S2CID 10132718.
7. Crosson, Elizabeth; Farhi, Edward; Cedric Yen-Yu Lin; Lin, Han-Hsuan; Shor, Peter (2014). "Different Strategies for Optimization Using the Quantum Adiabatic Algorithm". arXiv:1401.7320 [quant-ph].
8. Muthukrishnan, Siddharth; Albash, Tameem; Lidar, Daniel A. (2015). "When Diabatic Trumps Adiabatic in Quantum Optimization". arXiv:1505.01249 [quant-ph].
9. Brooke, J.; Bitko, D.; Rosenbaum, T. F.; Aeppli, G. (1999). "Quantum annealing of a disordered magnet". Science. 284 (5415): 779–81. arXiv:cond-mat/0105238. Bibcode:1999Sci...284..779B. doi:10.1126/science.284.5415.779. PMID 10221904. S2CID 37564720.
10. Sinitsyn, N.; Yan, B. (2023). "Topologically protected Grover's oracle for the Partition Problem". arXiv:2304.10488 [quant-ph].
11. Morita, Satoshi; Nishimori, Hidetoshi (2008). "Mathematical foundation of quantum annealing". Journal of Mathematical Physics. 49 (12): 125210. arXiv:0806.1859. Bibcode:2008JMP....49l5210M. doi:10.1063/1.2995837. S2CID 13992889.
12. Santoro, Giuseppe E. & Tosatti, Erio (18 August 2006). "Optimization using quantum mechanics: quantum annealing through adiabatic evolution". Journal of Physics A. 39 (36): R393–R431. Bibcode:2006JPhA...39R.393S. doi:10.1088/0305-4470/39/36/R01. S2CID 116931586.
13. Heim, B.; Rønnow, T. F.; Isakov, S. V.; Troyer, M. (2015). "Quantum versus classical annealing of Ising spin glasses". Science. 348 (6231): 215–217. arXiv:1411.5693. Bibcode:2015Sci...348..215H. doi:10.1126/science.aaa4170. PMID 25765071.
14. Yan, B.; Sinitsyn, N. A. (2022). "Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian". Nature Communications. 13 (1): 2212. arXiv:2110.12354. Bibcode:2022NatCo..13.2212Y. doi:10.1038/s41467-022-29887-0. PMC 9038765. PMID 35468917. S2CID 248389790.
15. "Local Maxima and Minima, and, Absolute Maxima and Minima". Mathonline.
16. Das, A.; Chakrabarti, B. K. & Stinchcombe, R. B. (2005). "Quantum annealing in a kinetically constrained system". Phys. Rev. E. 72 (2): 026701. arXiv:cond-mat/0502167. Bibcode:2005PhRvE..72b6701D. doi:10.1103/PhysRevE.72.026701. PMID 16196745. S2CID 16466621. Archived from the original on 2014-01-13.
17. See e.g., Mukherjee, S. & Chakrabarti, B. K. (2015). "Multivariable Optimization: Quantum Annealing & Computation". Eur. Phys. J. 224 (1): 17–24. arXiv:1408.3262. Bibcode:2015EPJST.224...17M. doi:10.1140/epjst/e2015-02339-y. S2CID 118525494.
18. Das, A.; Chakrabarti, B. K. (2008). "Quantum Annealing and Analog Quantum Computation". Rev. Mod. Phys. 80 (3): 1061–1081. arXiv:0801.2193. Bibcode:2008RvMP...80.1061D. CiteSeerX 10.1.1.563.9990. doi:10.1103/RevModPhys.80.1061. S2CID 14255125.
19. Li, F.; Chernyak, V. Y. & Sinitsyn, N. A. (2018). "Quantum annealing and thermalization: insights from integrability". Physical Review Letters. 121 (19): 190601. arXiv:1804.00371. Bibcode:2018arXiv180400371L. doi:10.1103/PhysRevLett.121.190601. PMID 30468584. S2CID 53594139.
20. Yan, B.; Sinitsyn, N. A. (2022). "Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian". Nature Communications. 13 (1): 2212. arXiv:2110.12354. Bibcode:2022NatCo..13.2212Y. doi:10.1038/s41467-022-29887-0. PMC 9038765. PMID 35468917. S2CID 248389790.
21. Brooke, J.; Bitko, D.; Rosenbaum, T. F. & Aeppli, G. (30 April 1999). "Quantum Annealing of a Disordered Magnet". Science. 284 (5415): 779–781. Bibcode:1999Sci...284..779B. doi:10.1126/science.284.5415.779. PMID 10221904. S2CID 37564720.
22. Johnson, M. W.; Amin, M. H. S.; Gildert, S.; et al. (2011). "Quantum annealing with manufactured spins". Nature. 473 (7346): 194–8. Bibcode:2011Natur.473..194J. doi:10.1038/nature10012. PMID 21562559. S2CID 205224761.
23. "Learning to program the D-Wave One". D-Wave Systems blog. Archived from the original on July 23, 2011. Retrieved 11 May 2011.
24. "D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". D-Wave. 2011-05-25. Archived from the original on July 23, 2011. Retrieved 2011-05-30.
25. Jones, N. (2013). "Google and NASA snap up quantum computer". Nature News. doi:10.1038/nature.2013.12999. S2CID 57405432.
26. Smelyanskiy, Vadim N.; Rieffel, Eleanor G.; Knysh, Sergey I.; Williams, Colin P.; Johnson, Mark W.; Thom, Murray C.; Macready, William G.; Pudenz, Kristen L. (2012). "A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration". arXiv:1204.2821 [quant-ph].
27. Boixo, S.; Rønnow, T. F.; Isakov, S. V.; Wang, Z.; Wecker, D.; Lidar, D. A.; Martinis, J. M.; Troyer, M. (2014). "Evidence for quantum annealing with more than one hundred qubits". Nature Physics. 10 (3): 218–224. arXiv:1304.4595. Bibcode:2014NatPh..10..218B. doi:10.1038/nphys2900. S2CID 8031023.
28. "D-Wave Systems Building Quantum Application Ecosystem, Announces Partnerships with DNA-SEQ Alliance and 1QBit". D-Wave Systems. Retrieved 22 June 2014.
29. "1QBit Research". 1QBit. Archived from the original on 19 June 2014. Retrieved 22 June 2014.
30. Lanting, T.; Przybysz, A. J.; Smirnov, A. Yu.; Spedalieri, F. M.; et al. (2014-05-29). "Entanglement in a quantum annealing processor". Physical Review X. 4 (2): 021041. arXiv:1401.3500. Bibcode:2014PhRvX...4b1041L. doi:10.1103/PhysRevX.4.021041. S2CID 19235104.
31. Helmut Katzgraber, quoted in (Cho 2014).
32. Cho, Adrian (20 June 2014). "Quantum or not, controversial computer yields no speedup". Science. 344 (6190): 1330–1331. Bibcode:2014Sci...344.1330C. doi:10.1126/science.344.6190.1330. PMID 24948715.
33. Amin, Mohammad H. (2015). "Searching for quantum speedup in quasistatic quantum annealers". Physical Review A. 92 (5): 052323. arXiv:1503.04216. Bibcode:2015PhRvA..92e2323A. doi:10.1103/PhysRevA.92.052323. S2CID 66770023.
34. Steiger, Damian; Heim, Bettina; Rønnow, Troels; Troyer, Matthias (October 22, 2015). Huckridge, David A.; Ebert, Reinhard; Gruneisen, Mark T.; Dusek, Miloslav; Rarity, John G. (eds.). "Performance of quantum annealing hardware". Proceedings of SPIE (Electro-Optical and Infrared Systems: Technology and Applications XII; and Quantum Information Science and Technology). Electro-Optical and Infrared Systems: Technology and Applications XII; and Quantum Information Science and Technology. 9648: 964816. Bibcode:2015SPIE.9648E..16S. doi:10.1117/12.2202661. S2CID 57916974.
35. "When can Quantum Annealing win?". Research Blog. 8 December 2015. Retrieved 2016-01-21.
36. D-Wave Systems (2021-10-05). "D-Wave's Next-Generation Roadmap: Bringing Clarity to Practical Quantum Computing". Medium. Retrieved 2021-11-12.
37. Venegas-Andraca, Salvador E.; Cruz-Santos, William; McGeoch, Catherine; Lanzagorta, Marco (2018). "A cross-disciplinary introduction to quantum annealing-based algorithms". Contemporary Physics. 59 (2): 174–196. arXiv:1803.03372. Bibcode:2018ConPh..59..174V. doi:10.1080/00107514.2018.1450720. S2CID 118974781.
38. Ajagekar, Akshay; Humble, Travis; You, Fengqi (2020-01-04). "Quantum computing based hybrid solution strategies for large-scale discrete-continuous optimization problems". Computers & Chemical Engineering. 132: 106630. arXiv:1910.13045. doi:10.1016/j.compchemeng.2019.106630. ISSN 0098-1354.
39. Wierzbiński, M.; Falo-Roget, J.; Crimi, A. (2023). "Community detection in brain connectomes with hybrid quantum computing". Scientific Reports. 13 (1): 3446. Bibcode:2023NatSR..13.3446W. doi:10.1038/s41598-023-30579-y. PMC 9977923. PMID 36859591. S2CID 257236235.
Further reading
• Bapst, V.; Foini, L.; Krzakala, F.; Semerjian, G.; Zamponi, F. (2013). "The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective". Physics Reports. 523 (3): 127–205. arXiv:1210.0811. Bibcode:2013PhR...523..127B. doi:10.1016/j.physrep.2012.10.002. S2CID 19019744.
• Chandra, Anjan K.; Das, Arnab & Chakrabarti, Bikas K., eds. (2010). Quantum Quenching, Annealing and Computation. Lecture Note in Physics. Vol. 802. Heidelberg: Springer. ISBN 978-3-64211-469-4.
• Das, Arnab & Chakrabarti, Bikas K., eds. (2005). Quantum Annealing and Related Optimization Methods. Lecture Note in Physics. Vol. 679. Heidelberg: Springer. ISBN 978-3-54027-987-7.
• Dutta, A.; Aeppli, G.; Chakrabarti, B. K.; Divakaran, U.; Rosenbaum, T.F. & Sen, D. (2015). Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information. Cambridge & Delhi: Cambridge University Press. ISBN 978-1-10706-879-7.
• Li, Fuxiang; Chernyak, V. Y.; Sinitsyn, N. A. (2013). "Quantum Annealing and Computation: A Brief Documentary Note". Science and Culture. 79: 485–500. arXiv:1310.1339. Bibcode:2013arXiv1310.1339G..
• Suzuki, S.; Inoue, J.-I. & Chakrabarti, B. K. (2013). "Chapter 8 on Quantum Annealing". Quantum Ising Phases & Transitions in Transverse Ising Models (2nd ed.). Heidelberg: Springer. ISBN 978-3-64233-038-4.
• Tanaka, S.; Tamura, R. & Chakrabarti, B. K. (2017). Quantum Spin Glasses, Annealing & Computation. Cambridge & Delhi: Cambridge University Press. ISBN 978-1-10711-319-0.
• "Quantum Annealing & Computation: Challenges & Perspectives". Philosophical Transactions A. Royal Society, London. 381. January 2023.
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Communication complexity
In theoretical computer science, communication complexity studies the amount of communication required to solve a problem when the input to the problem is distributed among two or more parties. The study of communication complexity was first introduced by Andrew Yao in 1979, while studying the problem of computation distributed among several machines.[1] The problem is usually stated as follows: two parties (traditionally called Alice and Bob) each receive a (potentially different) $n$-bit string $x$ and $y$. The goal is for Alice to compute the value of a certain function, $f(x,y)$, that depends on both $x$ and $y$, with the least amount of communication between them.
While Alice and Bob can always succeed by having Bob send his whole $n$-bit string to Alice (who then computes the function $f$), the idea here is to find clever ways of calculating $f$ with fewer than $n$ bits of communication. Note that, unlike in computational complexity theory, communication complexity is not concerned with the amount of computation performed by Alice or Bob, or the size of the memory used, as we generally assume nothing about the computational power of either Alice or Bob.
This abstract problem with two parties (called two-party communication complexity), and its general form with more than two parties, is relevant in many contexts. In VLSI circuit design, for example, one seeks to minimize energy used by decreasing the amount of electric signals passed between the different components during a distributed computation. The problem is also relevant in the study of data structures and in the optimization of computer networks. For surveys of the field, see the textbooks by Rao & Yehudayoff (2020) and Kushilevitz & Nisan (2006).
Formal definition
Let $f:X\times Y\rightarrow Z$ where we assume in the typical case that $X=Y=\{0,1\}^{n}$ and $Z=\{0,1\}$. Alice holds an $n$-bit string $x\in X$ while Bob holds an $n$-bit string $y\in Y$. By communicating to each other one bit at a time (adopting some communication protocol which is agreed upon in advance), Alice and Bob wish to compute the value of $f(x,y)$ such that at least one party knows the value at the end of the communication. At this point the answer can be communicated back so that at the cost of one extra bit, both parties will know the answer. The worst case communication complexity of this communication problem of computing $f$, denoted as $D(f)$, is then defined to be
$D(f)=$ minimum number of bits exchanged between Alice and Bob in the worst case.
As observed above, for any function $f:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$, we have $D(f)\leq n$. Using the above definition, it is useful to think of the function $f$ as a matrix $A$ (called the input matrix or communication matrix) where the rows are indexed by $x\in X$ and columns by $y\in Y$. The entries of the matrix are $A_{x,y}=f(x,y)$. Initially both Alice and Bob have a copy of the entire matrix $A$ (assuming the function $f$ is known to both parties). Then, the problem of computing the function value can be rephrased as "zeroing-in" on the corresponding matrix entry. This problem can be solved if either Alice or Bob knows both $x$ and $y$. At the start of communication, the number of choices for the value of the function on the inputs is the size of matrix, i.e. $2^{2n}$. Then, as and when each party communicates a bit to the other, the number of choices for the answer reduces as this eliminates a set of rows/columns resulting in a submatrix of $A$.
More formally, a set $R\subseteq X\times Y$ is called a (combinatorial) rectangle if whenever $(x_{1},y_{1})\in R$ and $(x_{2},y_{2})\in R$ then $(x_{1},y_{2})\in R$. Equivalently, $R$ is a combinatorial rectangle if it can be expressed as $R=M\times N$ for some $M\subseteq X$ and $N\subseteq Y$. Consider the case when $k$ bits are already exchanged between the parties. Now, for a particular $h\in \{0,1\}^{k}$, let us define a matrix
$T_{h}=\{(x,y):{\text{ the }}k{\text{-bits exchanged on input }}(x,y){\text{ is }}h\}$
Then, $T_{h}\subseteq X\times Y$, and it is not hard to show that $T_{h}$ is a combinatorial rectangle in $A$.
Example: $EQ$
We consider the case where Alice and Bob try to determine whether or not their input strings are equal. Formally, define the Equality function, denoted $EQ:\{0,1\}^{n}\times \{0,1\}^{n}\rightarrow \{0,1\}$, by $EQ(x,y)=1$ if $x=y$. As we demonstrate below, any deterministic communication protocol solving $EQ$ requires $n$ bits of communication in the worst case. As a warm-up example, consider the simple case of $x,y\in \{0,1\}^{3}$. The equality function in this case can be represented by the matrix below. The rows represent all the possibilities of $x$, the columns those of $y$.
EQ 000 001 010 011 100 101 110 111
000 1 0 0 0 0 0 0 0
001 0 1 0 0 0 0 0 0
010 0 0 1 0 0 0 0 0
011 0 0 0 1 0 0 0 0
100 0 0 0 0 1 0 0 0
101 0 0 0 0 0 1 0 0
110 0 0 0 0 0 0 1 0
111 0 0 0 0 0 0 0 1
As you can see, the function only evaluates to 1 when $x$ equals $y$ (i.e., on the diagonal). It is also fairly easy to see how communicating a single bit divides your possibilities in half. If you know that the first bit of $y$ is 1, you only need to consider half of the columns (where $y$ can equal 100, 101, 110, or 111).
Theorem: $D(EQ)=n$
Proof. Assume that $D(EQ)\leq n-1$. This means that there exists $x\neq x'$ such that $(x,x)$ and $(x',x')$ have the same communication transcript $h$. Since this transcript defines a rectangle, $f(x,x')$ must also be 1. By definition $x\neq x'$ and we know that equality is only true for $(a,b)$ when $a=b$. This yields a contradiction.
This technique of proving deterministic communication lower bounds is called the fooling set technique.[2]
Randomized communication complexity
In the above definition, we are concerned with the number of bits that must be deterministically transmitted between two parties. If both the parties are given access to a random number generator, can they determine the value of $f$ with much less information exchanged? Yao, in his seminal paper[1] answers this question by defining randomized communication complexity.
A randomized protocol $R$ for a function $f$ has two-sided error.
$\Pr[R(x,y)=0]>{\frac {2}{3}},{\textrm {if}}\,f(x,y)=0$
$\Pr[R(x,y)=1]>{\frac {2}{3}},{\textrm {if}}\,f(x,y)=1$
A randomized protocol is a deterministic protocol that uses an extra random string in addition to its normal input. There are two models for this: a public string is a random string that is known by both parties beforehand, while a private string is generated by one party and must be communicated to the other party. A theorem presented below shows that any public string protocol can be simulated by a private string protocol that uses O(log n) additional bits compared to the original.
Note that in the probability inequalities above, the outcome of the protocol is understood to depend only on the random string; both strings x and y remain fixed. In other words, if R(x,y) yields g(x,y,r) when using random string r, then g(x,y,r) = f(x,y) for at least 2/3 of all choices for the string r.
The randomized complexity is simply defined as the number of bits exchanged in such a protocol.
Note that it is also possible to define a randomized protocol with one-sided error, and the complexity is defined similarly.
Example: EQ
Returning to the previous example of EQ, if certainty is not required, Alice and Bob can check for equality using only $O(\log n)$ messages. Consider the following protocol: Assume that Alice and Bob both have access to the same random string $z\in \{0,1\}^{n}$. Alice computes $z\cdot x$ and sends this bit (call it b) to Bob. (The $(\cdot )$ is the dot product in GF(2).) Then Bob compares b to $z\cdot y$. If they are the same, then Bob accepts, saying x equals y. Otherwise, he rejects.
Clearly, if $x=y$, then $z\cdot x=z\cdot y$, so $Prob_{z}[Accept]=1$. If x does not equal y, it is still possible that $z\cdot x=z\cdot y$, which would give Bob the wrong answer. How does this happen?
If x and y are not equal, they must differ in some locations:
${\begin{cases}x=c_{1}c_{2}\ldots p\ldots p'\ldots x_{n}\\y=c_{1}c_{2}\ldots q\ldots q'\ldots y_{n}\\z=z_{1}z_{2}\ldots z_{i}\ldots z_{j}\ldots z_{n}\end{cases}}$
Where x and y agree, $z_{i}*x_{i}=z_{i}*c_{i}=z_{i}*y_{i}$ so those terms affect the dot products equally. We can safely ignore those terms and look only at where x and y differ. Furthermore, we can swap the bits $x_{i}$ and $y_{i}$ without changing whether or not the dot products are equal. This means we can swap bits so that x contains only zeros and y contains only ones:
${\begin{cases}x'=00\ldots 0\\y'=11\ldots 1\\z'=z_{1}z_{2}\ldots z_{n'}\end{cases}}$
Note that $z'\cdot x'=0$ and $z'\cdot y'=\Sigma _{i}z'_{i}$. Now, the question becomes: for some random string $z'$, what is the probability that $\Sigma _{i}z'_{i}=0$? Since each $z'_{i}$ is equally likely to be 0 or 1, this probability is just $1/2$. Thus, when x does not equal y, $Prob_{z}[Accept]=1/2$. The algorithm can be repeated many times to increase its accuracy. This fits the requirements for a randomized communication algorithm.
This shows that if Alice and Bob share a random string of length n, they can send one bit to each other to compute $EQ(x,y)$. In the next section, it is shown that Alice and Bob can exchange only $O(\log n)$ bits that are as good as sharing a random string of length n. Once that is shown, it follows that EQ can be computed in $O(\log n)$ messages.
Example: GH
For yet another example of randomized communication complexity, we turn to an example known as the gap-Hamming problem (abbreviated GH). Formally, Alice and Bob both maintain binary messages, $x,y\in \{-1,+1\}^{n}$ and would like to determine if the strings are very similar or if they are not very similar. In particular, they would like to find a communication protocol requiring the transmission of as few bits as possible to compute the following partial Boolean function,
${\text{GH}}_{n}(x,y):={\begin{cases}-1&\langle x,y\rangle \leq {\sqrt {n}}\\+1&\langle x,y\rangle \geq {\sqrt {n}}.\end{cases}}$
Clearly, they must communicate all their bits if the protocol is to be deterministic (this is because, if there is a deterministic, strict subset of indices that Alice and Bob relay to one another, then imagine having a pair of strings that on that set disagree in ${\sqrt {n}}-1$ positions. If another disagreement occurs in any position that is not relayed, then this affects the result of ${\text{GH}}_{n}(x,y)$, and hence would result in an incorrect procedure.
A natural question one then asks is, if we're permitted to err $1/3$ of the time (over random instances $x,y$ drawn uniformly at random from $\{-1,+1\}^{n}$), then can we get away with a protocol with fewer bits? It turns out that the answer somewhat surprisingly is no, due to a result of Chakrabarti and Regev in 2012: they show that for random instances, any procedure which is correct at least $2/3$ of the time must send $\Omega (n)$ bits worth of communication, which is to say essentially all of them.
Public coins versus private coins
It is easier to create random protocols when both parties have access to the same random string (shared string protocol). It is still possible to use these protocols even when the two parties don't share a random string (private string protocol) with a small communication cost. Any shared string random protocol using any number of random string can be simulated by a private string protocol that uses an extra O(log n) bits.
Intuitively, we can find some set of strings that has enough randomness in it to run the random protocol with only a small increase in error. This set can be shared beforehand, and instead of drawing a random string, Alice and Bob need only agree on which string to choose from the shared set. This set is small enough that the choice can be communicated efficiently. A formal proof follows.
Consider some random protocol P with a maximum error rate of 0.1. Let $R$ be $100n$ strings of length n, numbered $r_{1},r_{2},\dots ,r_{100n}$. Given such an $R$, define a new protocol $P'_{R}$ which randomly picks some $r_{i}$ and then runs P using $r_{i}$ as the shared random string. It takes O(log 100n) = O(log n) bits to communicate the choice of $r_{i}$.
Let us define $p(x,y)$ and $p'_{R}(x,y)$ to be the probabilities that $P$ and $P'_{R}$ compute the correct value for the input $(x,y)$.
For a fixed $(x,y)$, we can use Hoeffding's inequality to get the following equation:
$\Pr _{R}[|p'_{R}(x,y)-p(x,y)|\geq 0.1]\leq 2\exp(-2(0.1)^{2}\cdot 100n)<2^{-2n}$
Thus when we don't have $(x,y)$ fixed:
$\Pr _{R}[\exists (x,y):\,|p'_{R}(x,y)-p(x,y)|\geq 0.1]\leq \sum _{(x,y)}\Pr _{R}[|p'_{R}(x,y)-p(x,y)|\geq 0.1]<\sum _{(x,y)}2^{-2n}=1$
The last equality above holds because there are $2^{2n}$ different pairs $(x,y)$. Since the probability does not equal 1, there is some $R_{0}$ so that for all $(x,y)$:
$|p'_{R_{0}}(x,y)-p(x,y)|<0.1$
Since $P$ has at most 0.1 error probability, $P'_{R_{0}}$ can have at most 0.2 error probability.
Collapse of Randomized Communication Complexity
Let's say we additionally allow Alice and Bob to share some resource, for example a pair of entangle particles. Using that ressource, Alice and Bob can correlate their information and thus try to 'collapse' (or 'trivialize') communication complexity in the following sense.
Definition. A resource $R$ is said to be "collapsing" if, using that resource $R$, only one bit of classical communication is enough for Alice to know the evaluation $f(x,y)$ in the worst case scenario for any Boolean function $f$.
The surprising fact of a collapse of communication complexity is that the function $f$ can have arbitrarily large entry size, but still the number of communication bit is constant to a single one.
Some resources are shown to be non-collapsing, such as quantum correlations [3] or more generally almost-quantum correlations,[4] whereas on the contrary some other resources are shown to collapse randomized communication complexity, such as the PR-box,[5] or some noisy PR-boxes satisfying some conditions.[6][7][8]
Quantum communication complexity
Quantum communication complexity tries to quantify the communication reduction possible by using quantum effects during a distributed computation.
At least three quantum generalizations of communication complexity have been proposed; for a survey see the suggested text by G. Brassard.
The first one is the qubit-communication model, where the parties can use quantum communication instead of classical communication, for example by exchanging photons through an optical fiber.
In a second model the communication is still performed with classical bits, but the parties are allowed to manipulate an unlimited supply of quantum entangled states as part of their protocols. By doing measurements on their entangled states, the parties can save on classical communication during a distributed computation.
The third model involves access to previously shared entanglement in addition to qubit communication, and is the least explored of the three quantum models.
Nondeterministic communication complexity
In nondeterministic communication complexity, Alice and Bob have access to an oracle. After receiving the oracle's word, the parties communicate to deduce $f(x,y)$. The nondeterministic communication complexity is then the maximum over all pairs $(x,y)$ over the sum of number of bits exchanged and the coding length of the oracle word.
Viewed differently, this amounts to covering all 1-entries of the 0/1-matrix by combinatorial 1-rectangles (i.e., non-contiguous, non-convex submatrices, whose entries are all one (see Kushilevitz and Nisan or Dietzfelbinger et al.)). The nondeterministic communication complexity is the binary logarithm of the rectangle covering number of the matrix: the minimum number of combinatorial 1-rectangles required to cover all 1-entries of the matrix, without covering any 0-entries.
Nondeterministic communication complexity occurs as a means to obtaining lower bounds for deterministic communication complexity (see Dietzfelbinger et al.), but also in the theory of nonnegative matrices, where it gives a lower bound on the nonnegative rank of a nonnegative matrix.[9]
Unbounded-error communication complexity
In the unbounded-error setting, Alice and Bob have access to a private coin and their own inputs $(x,y)$. In this setting, Alice succeeds if she responds with the correct value of $f(x,y)$ with probability strictly greater than 1/2. In other words, if Alice's responses have any non-zero correlation to the true value of $f(x,y)$, then the protocol is considered valid.
Note that the requirement that the coin is private is essential. In particular, if the number of public bits shared between Alice and Bob are not counted against the communication complexity, it is easy to argue that computing any function has $O(1)$ communication complexity.[10] On the other hand, both models are equivalent if the number of public bits used by Alice and Bob is counted against the protocol's total communication.[11]
Though subtle, lower bounds on this model are extremely strong. More specifically, it is clear that any bound on problems of this class immediately imply equivalent bounds on problems in the deterministic model and the private and public coin models, but such bounds also hold immediately for nondeterministic communication models and quantum communication models.[12]
Forster[13] was the first to prove explicit lower bounds for this class, showing that computing the inner product $\langle x,y\rangle $ requires at least $\Omega (n)$ bits of communication, though an earlier result of Alon, Frankl, and Rödl proved that the communication complexity for almost all Boolean functions $f:\{0,1\}^{n}\times \{0,1\}^{n}\to \{0,1\}$ is $\Omega (n)$.[14]
Open problems
Considering a 0 or 1 input matrix $M_{f}=[f(x,y)]_{x,y\in \{0,1\}^{n}}$, the minimum number of bits exchanged to compute $f$ deterministically in the worst case, $D(f)$, is known to be bounded from below by the logarithm of the rank of the matrix $M_{f}$. The log rank conjecture proposes that the communication complexity, $D(f)$, is bounded from above by a constant power of the logarithm of the rank of $M_{f}$. Since D(f) is bounded from above and below by polynomials of log rank$(M_{f})$, we can say D(f) is polynomially related to log rank$(M_{f})$. Since the rank of a matrix is polynomial time computable in the size of the matrix, such an upper bound would allow the matrix's communication complexity to be approximated in polynomial time. Note, however, that the size of the matrix itself is exponential in the size of the input.
For a randomized protocol, the number of bits exchanged in the worst case, R(f), was conjectured to be polynomially related to the following formula:
$\log \min({\textrm {rank}}(M'_{f}):M'_{f}\in \mathbb {R} ^{2^{n}\times 2^{n}},(M_{f}-M'_{f})_{\infty }\leq 1/3).$
Such log rank conjectures are valuable because they reduce the question of a matrix's communication complexity to a question of linearly independent rows (columns) of the matrix. This particular version, called the Log-Approximate-Rank Conjecture, was recently refuted by Chattopadhyay, Mande and Sherif (2019)[15] using a surprisingly simple counter-example. This reveals that the essence of the communication complexity problem, for example in the EQ case above, is figuring out where in the matrix the inputs are, in order to find out if they're equivalent.
Applications
Lower bounds in communication complexity can be used to prove lower bounds in decision tree complexity, VLSI circuits, data structures, streaming algorithms, space–time tradeoffs for Turing machines and more.[2]
See also
• Gap-Hamming problem
Notes
1. Yao, A. C. (1979), "Some Complexity Questions Related to Distributive Computing", Proc. Of 11th STOC, 14: 209–213
2. Kushilevitz, Eyal; Nisan, Noam (1997). Communication Complexity. Cambridge University Press. ISBN 978-0-521-56067-2.
3. R. Cleve, W. van Dam, M. Nielsen, and A. Tapp, Quantum Computing and Quantum Communications (Springer Berlin Heidelberg, Berlin, Heidelberg, 1999) pp. 61–74.
4. M. Navascués, Y. Guryanova, M. J. Hoban, and A. Acín, Nature Communications 6, 6288 (2015).
5. W. van Dam, Nonlocality & Communication Complexity, Ph.d. thesis, University of Oxford (1999).
6. G. Brassard, H. Buhrman, N. Linden, A. A. M ́ethot, A. Tapp, and F. Unger, Phys. Rev. Lett. 96, 250401 (2006).
7. N. Brunner and P. Skrzypczyk, Phys. Rev. Lett. 102, 160403 (2009).
8. P. Botteron, A. Broadbent and M.-O. Proulx, arXiv:2302.00488.
9. Yannakakis, M. (1991). "Expressing combinatorial optimization problems by linear programs". J. Comput. Syst. Sci. 43 (3): 441–466. doi:10.1016/0022-0000(91)90024-y.
10. Lovett, Shachar, CSE 291: Communication Complexity, Winter 2019 Unbounded-error protocols (PDF), retrieved June 9, 2019
11. Göös, Mika; Pitassi, Toniann; Watson, Thomas (2018-06-01). "The Landscape of Communication Complexity Classes". Computational Complexity. 27 (2): 245–304. doi:10.1007/s00037-018-0166-6. ISSN 1420-8954. S2CID 4333231.
12. Sherstov, Alexander A. (October 2008). "The Unbounded-Error Communication Complexity of Symmetric Functions". 2008 49th Annual IEEE Symposium on Foundations of Computer Science. pp. 384–393. doi:10.1109/focs.2008.20. ISBN 978-0-7695-3436-7. S2CID 9072527.
13. Forster, Jürgen (2002). "A linear lower bound on the unbounded error probabilistic communication complexity". Journal of Computer and System Sciences. 65 (4): 612–625. doi:10.1016/S0022-0000(02)00019-3.
14. Alon, N.; Frankl, P.; Rodl, V. (October 1985). "Geometrical realization of set systems and probabilistic communication complexity". 26th Annual Symposium on Foundations of Computer Science (SFCS 1985). Portland, OR, USA: IEEE. pp. 277–280. CiteSeerX 10.1.1.300.9711. doi:10.1109/SFCS.1985.30. ISBN 9780818606441. S2CID 8416636.
15. Chattopadhyay, Arkadev; Mande, Nikhil S.; Sherif, Suhail (2019). "The Log-Approximate-Rank Conjecture is False". 2019, Proceeding of the 51st Annual ACM Symposium on Theory of Computing: 42-53.https://doi.org/10.1145/3313276.3316353
References
• Rao, Anup; Yehudayoff, Amir (2020). Communication complexity and applications. Cambridge: Cambridge University Press. ISBN 9781108671644.
• Kushilevitz, Eyal; Nisan, Noam (2006). Communication complexity. Cambridge: Cambridge University Press. ISBN 978-0-521-02983-4. OCLC 70764786.
• Brassard, G. Quantum communication complexity: a survey. https://arxiv.org/abs/quant-ph/0101005
• Dietzfelbinger, M., J. Hromkovic, J., and G. Schnitger, "A comparison of two lower-bound methods for communication complexity", Theoret. Comput. Sci. 168, 1996. 39-51.
• Raz, Ran. "Circuit and Communication Complexity." In Computational Complexity Theory. Steven Rudich and Avi Wigderson, eds. American Mathematical Society Institute for Advanced Study, 2004. 129-137.
• A. C. Yao, "Some Complexity Questions Related to Distributed Computing", Proc. of 11th STOC, pp. 209–213, 1979. 14
• I. Newman, Private vs. Common Random Bits in Communication Complexity, Information Processing Letters 39, 1991, pp. 67–71.
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Wikipedia
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Quantum complex network
Quantum complex networks are complex networks whose nodes are quantum computing devices.[1][2] Quantum mechanics has been used to create secure quantum communications channels that are protected from hacking.[3][4] Quantum communications offer the potential for secure enterprise-scale solutions.[5][2][6]
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Motivation
In theory, it is possible to take advantage of quantum mechanics to create secure communications using features such as quantum key distribution is an application of quantum cryptography that enables secure communications[3] Quantum teleportation can transfer data at a higher rate than classical channels.[4]
History
Successful quantum teleportation experiments in 1998.[7] Prototypical quantum communication networks arrived in 2004.[8] Large scale communication networks tend to have non-trivial topologies and characteristics, such as small world effect, community structure, or scale-free.[6]
Concepts
Qubits
In quantum information theory, qubits are analogous to bits in classical systems. A qubit is a quantum object that, when measured, can be found to be in one of only two states, and that is used to transmit information.[3] Photon polarization or nuclear spin are examples of binary phenomena that can be used as qubits.[3]
Entanglement
Quantum entanglement is a physical phenomenon characterized by correlation between the quantum states of two or more physically separate qubits.[3] Maximally entangled states are those that maximize the entropy of entanglement.[9][10] In the context of quantum communication, entangled qubits are used as a quantum channel.[3]
Bell measurement
Bell measurement is a kind of joint quantum-mechanical measurement of two qubits such that, after the measurement, the two qubits are maximally entangled.[3][10]
Entanglement swapping
Entanglement swapping is a strategy used in the study of quantum networks that allows connections in the network to change.[1][11] For example, given 4 qubits, A, B, C and D, such that qubits C and D belong to the same station, while A and C belong to two different stations, and where qubit A is entangled with qubit C and qubit B is entangled with qubit D. Performing a Bell measurement for qubits A and B, entangles qubits A and B. It is also possible to entangle qubits C and D, despite the fact that these two qubits never interact directly with each other. Following this process, the entanglement between qubits A and C, and qubits B and D are lost. This strategy can be used to define network topology.[1][11][12]
Network structure
While models for quantum complex networks are not of identical structure, usually a node represents a set of qubits in the same station (where operations like Bell measurements and entanglement swapping can be applied) and an edge between node $i$ and $j$ means that a qubit in node $i$ is entangled to a qubit in node $j$, although those two qubits are in different places and so cannot physically interact.[1][11] Quantum networks where the links are interaction terms instead of entanglement are also of interest[13]
Notation
Each node in the network contains a set of qubits in different states. To represent the quantum state of these qubits, it is convenient to use Dirac notation and represent the two possible states of each qubit as $|0\rangle $ and $|1\rangle $.[1][11] In this notation, two particles are entangled if the joint wave function, $|\psi _{ij}\rangle $, cannot be decomposed as[3][10]
$|\psi _{ij}\rangle =|\phi \rangle _{i}\otimes |\phi \rangle _{j},$
where $|\phi \rangle _{i}$ represents the quantum state of the qubit at node i and $|\phi \rangle _{j}$ represents the quantum state of the qubit at node j.
Another important concept is maximally entangled states. The four states (the Bell states) that maximize the entropy of entanglement between two qubits can be written as follows:[3][10]
$|\Phi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j}),$
$|\Phi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}-|1\rangle _{i}\otimes |1\rangle _{j}),$
$|\Psi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}+|1\rangle _{i}\otimes |0\rangle _{j}),$
$|\Psi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}-|1\rangle _{i}\otimes |0\rangle _{j}).$
Models
Quantum random networks
The quantum random network model proposed by Perseguers et al. (2009)[1] can be thought of as a quantum version of the Erdős–Rényi model. In this model, each node contains $N-1$ qubits, one for each other node. The degree of entanglement between a pair of nodes, represented by $p$, plays a similar role to the parameter $p$ in the Erdős–Rényi model in which two nodes form a connection with probability $p$, whereas in the context of quantum random networks, $p$ refers to the probability of converting an entangled pair of qubits to a maximally entangled state using only local operations and classical communication.[14]
Using Dirac notation, a pair of entangled qubits connecting the nodes $i$ and $j$ is represented as
$|\psi _{ij}\rangle ={\sqrt {1-p/2}}|0\rangle _{i}\otimes |0\rangle _{j}+{\sqrt {p/2}}|1\rangle _{i}\otimes |1\rangle _{j},$
For $p=0$, the two qubits are not entangled:
$|\psi _{ij}\rangle =|0\rangle _{i}\otimes |0\rangle _{j},$
and for $p=1$, we obtain the maximally entangled state:
$|\psi _{ij}\rangle ={\sqrt {1/2}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j})$.
For intermediate values of $p$, $0<p<1$, any entangled state is, with probability $p$, successfully converted to the maximally entangled state using LOCC operations.[14]
One feature that distinguishes this model from its classical analogue is the fact that, in quantum random networks, links are only truly established after they are measured, and it is possible to exploit this fact to shape the final state of the network. For an initial quantum complex network with an infinite number of nodes, Perseguers et al.[1] showed that, the right measurements and entanglement swapping, make it possible to collapse the initial network to a network containing any finite subgraph, provided that $p$ scales with $N$ as $ p\sim N^{Z}$, where $Z\geq -2$. This result is contrary to classical graph theory, where the type of subgraphs contained in a network is bounded by the value of $z$.[15]
Entanglement percolation
Entanglement percolation models attempt to determine whether a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find the best strategies to create such connections.[11][16]
Cirac et al. (2007)[16] applied a model to complex networks by Cuquet et al. (2009),[11] in which nodes are distributed in a lattice[16] or in a complex network,[11] and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangled qubit pair with probability $p$. We can think of maximally entangled qubits as the true links between nodes. In classical percolation theory, with a probability $p$ that two nodes are connected, $p$ has a critical value (denoted by $p_{c}$), so that if $p>p_{c}$ a path between two randomly selected nodes exists with a finite probability, and for $p<p_{c}$ the probability of such a path existing is asymptotically zero.[17] $p_{c}$ depends only on the network topology.[17]
A similar phenomenon was found in the model proposed by Cirac et al. (2007),[16] where the probability of forming a maximally entangled state between two randomly selected nodes is zero if $p<p_{c}$ and finite if $p>p_{c}$. The main difference between classical and entangled percolation is that, in quantum networks, it is possible to change the links in the network, in a way changing the effective topology of the network. As a result, $p_{c}$ depends on the strategy used to convert partially entangled qubits to maximally connected qubits.[11][16] With a naïve approach, $p_{c}$ for a quantum network is equal to $p_{c}$ for a classic network with the same topology.[16] Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower $p_{c}$ both in regular lattices[16] and complex networks.[11]
See also
• Erdős–Rényi model
• Gradient network
• Network dynamics
• Network topology
• Quantum key distribution
• Quantum teleportation
References
1. Perseguers, S.; Lewenstein, M.; Acín, A.; Cirac, J. I. (16 May 2010) [19 July 2009]. "Quantum random networks" [Quantum complex networks]. Nature Physics. 6 (7): 539–543. arXiv:0907.3283. Bibcode:2010NatPh...6..539P. doi:10.1038/nphys1665. S2CID 119181158.
2. Cuquet, Martí; Calsamiglia, John (2009). "Entanglement Percolation in Quantum Complex Networks". Physical Review Letters. 103 (24): 240503. arXiv:0906.2977. Bibcode:2009PhRvL.103x0503C. doi:10.1103/physrevlett.103.240503. PMID 20366190. S2CID 19441960.
3. Nielsen, Michael A.; Chuang, Isaac L. (1 January 2004). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 978-1-107-00217-3.
4. Takeda, Shuntaro; Mizuta, Takahiro; Fuwa, Maria; Loock, Peter van; Furusawa, Akira (14 August 2013). "Deterministic quantum teleportation of photonic quantum bits by a hybrid technique". Nature. 500 (7462): 315–318. arXiv:1402.4895. Bibcode:2013Natur.500..315T. doi:10.1038/nature12366. PMID 23955230. S2CID 4344887.
5. Huang, Liang; Lai, Ying C. (2011). "Cascading dynamics in complex quantum networks". Chaos: An Interdisciplinary Journal of Nonlinear Science. 21 (2): 025107. Bibcode:2011Chaos..21b5107H. doi:10.1063/1.3598453. PMID 21721785.
6. Dorogovtsev, S.N.; Mendes, J.F.F. (2003). Evolution of Networks: From biological networks to the Internet and WWW. Oxford University Press. ISBN 978-0-19-851590-6.
7. Boschi, D.; Branca, S.; De Martini, F.; Hardy, L.; Popescu, S. (1998). "Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels". Physical Review Letters. 80 (6): 1121–1125. arXiv:quant-ph/9710013. Bibcode:1998PhRvL..80.1121B. doi:10.1103/physrevlett.80.1121. S2CID 15020942.
8. Elliott, Chip; Colvin, Alexander; Pearson, David; Pikalo, Oleksiy; Schlafer, John; Yeh, Henry (17 March 2005). "Current status of the DARPA Quantum Network". arXiv:quant-ph/0503058. Bibcode:2005quant.ph..3058E. {{cite journal}}: Cite journal requires |journal= (help)
9. Eisert, J.; Cramer, M.; Plenio, M. B. (February 2010). "Colloquium: Area laws for the entanglement entropy". Reviews of Modern Physics. 82 (1): 277–306. arXiv:0808.3773. Bibcode:2010RvMP...82..277E. doi:10.1103/RevModPhys.82.277.
10. Chandra, Naresh; Ghosh, Rama (2013). Quantum Entanglement in Electron Optics: Generation, Characterization, and Applications. Springer Series on Atomic, Optical, and Plasma Physics. Vol. 67. Springer. p. 43. ISBN 978-3642240706.
11. Cuquet, M.; Calsamiglia, J. (10 December 2009) [6 June 2009]. "Entanglement percolation in quantum complex networks". Physical Review Letters. 103 (24): 240503. arXiv:0906.2977. Bibcode:2009PhRvL.103x0503C. doi:10.1103/physrevlett.103.240503. PMID 20366190. S2CID 19441960.
12. Coecke, Bob (2003). "The logic of entanglement" (RR-03-12). Department of Computer Science, University of Oxford. arXiv:quant-ph/0402014. Bibcode:2004quant.ph..2014C. {{cite journal}}: Cite journal requires |journal= (help)
13. Nokkala, Johannes (2018-12-01). "Quantum complex networks (Doctoral dissertation)". {{cite journal}}: Cite journal requires |journal= (help)
14. Werner, Reinhard F. (15 Oct 1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/physreva.40.4277. PMID 9902666.
15. Albert, Réka; Barabási, Albert L. (Jan 2002). "Statistical mechanics of complex networks". Reviews of Modern Physics. 74 (1): 47–97. arXiv:cond-mat/0106096. Bibcode:2002RvMP...74...47A. doi:10.1103/revmodphys.74.47. S2CID 60545.
16. Acin, Antonio; Cirac, J. Ignacio; Lewenstein, Maciej (25 February 2007). "Entanglement percolation in quantum networks". Nature Physics. 3 (4): 256–259. arXiv:quant-ph/0612167. Bibcode:2007NatPh...3..256A. doi:10.1038/nphys549. S2CID 118987352.
17. Stauffer, Dietrich; Aharony, Anthony (1994). Introduction to Percolation Theory (2nd ed.). CRC Press. ISBN 978-0-7484-0253-3.
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